six transiting planets and a chain of laplace resonances

Astronomy & Astrophysics manuscript no. output ©ESO 2020October 27, 2020

Six transiting planets and a chain of Laplace resonances inTOI-178

A. Leleu?1,2, Y. Alibert2, N. C. Hara?1, M. J. Hooton2, T. G. Wilson3, P. Robutel4, J.-B. Delisle1, J. Laskar4,S. Hoyer5, C. Lovis1, E. M. Bryant6,7, E. Ducrot8, J. Cabrera9, J. Acton10, V. Adibekyan11,12,13, R. Allart1,

C. Allende Prieto14,15, R. Alonso14,15, D. Alves16, D. R. Anderson6,7, D. Angerhausen17, G. AngladaEscudé18,19, J. Asquier20, D. Barrado21, S. C. C. Barros11,13, W. Baumjohann22, D. Bayliss6,7, M. Beck1, T.Beck2, A. Bekkelien1, W. Benz2,23, N. Billot1, A. Bonfanti22, X. Bonfils24, F. Bouchy1, V. Bourrier1, G.Boué4, A. Brandeker25, C. Broeg2, M. Buder26, A. Burdanov8,27, M. R. Burleigh10, T. Bárczy28, A. C.

Cameron3, S. Chamberlain10, S. Charnoz29, B. F. Cooke6,7, C. Corral Van Damme20, A. C. M. Correia30,4, S.Cristiani31, M. Damasso32, M. B. Davies33, M. Deleuil5, L. Delrez34,35,1, O. D. S. Demangeon11,13, B.-O.

Demory23, P. Di Marcantonio31, G. Di Persio36, X. Dumusque1, D. Ehrenreich1, A. Erikson9, P. Figueira11,37,A. Fortier2,23, L. Fossati22, M. Fridlund38,39, D. Futyan1, D. Gandolfi40, A. García Muñoz41, L. Garcia8, S.

Gill6,7, E. Gillen??42,43, M. Gillon34, M. R. Goad10, J.I. González Hernández14,15, M. Guedel44, M. N.Günther???45, J. Haldemann2, B. Henderson10, K. Heng23, A. E. Hogan10, E. Jehin46, J. S. Jenkins47,48, A.Jordán49,50, L. Kiss51, M. H. Kristiansen52,53, K. Lam9, B. Lavie1, A. Lecavelier des Etangs54, M. Lendl1, J.Lillo-Box21, G. Lo Curto37, D. Magrin55, C. J. A. P. Martins11,12, P. F. L. Maxted56, J. McCormac57, A.Mehner37, G. Micela58, P. Molaro31,59, M. Moyano60, C. A. Murray43, V. Nascimbeni55, N. J. Nunes61, G.Olofsson25, H. P. Osborn23,45, M. Oshagh14,15, R. Ottensamer62, I. Pagano63, E. Pallé64,65, P. P. Pedersen43,F. A. Pepe1, C.M. Persson39, G. Peter26, G. Piotto55,66, G. Polenta67, D. Pollacco68, E. Poretti69,70, F. J.Pozuelos34,35, F. Pozuelos8,46, D. Queloz1,43, R. Ragazzoni55, N. Rando20, F. Ratti20, H. Rauer9,41,71, L.

Raynard10, R. Rebolo14,15, C. Reimers62, I. Ribas18,19, N. C. Santos11,13, G. Scandariato63, J. Schneider72, D.Sebastian73, M. Sestovic23, A. E. Simon2, A. M. S. Smith9, S. G. Sousa11, A. Sozzetti32, M. Steller22, A.

Suárez Mascareño14,15, G. M. Szabó74,75, D. Ségransan1, N. Thomas2, S. Thompson43, R. H. Tilbrook10, A.Triaud73, S. Udry1, V. Van Grootel35, H. Venus26, F. Verrecchia67,76, J. I. Vines16, N. A. Walton77, R. G.

West6,7, P. J. Wheatley6,7, D. Wolter9, and M. R. Zapatero Osorio78

(Affiliations can be found after the references)

October 27, 2020

ABSTRACT

Determining the architecture of multi-planetary systems is one of the cornerstones of understanding planet formationand evolution. Among these, resonant systems are especially important as the fragility of their orbital configurationensure that no significant scattering or collisional event took place since the earliest formation phase, when the parentprotoplanetary disk was still present. In this context, TOI-178 has been the subject of particular attention as thefirst TESS observations hinted at the possible presence of a near 2:3:3 resonant chain. Here we report the results ofobservations from CHEOPS, ESPRESSO, NGTS, and SPECULOOS with the aim of deciphering the peculiar orbitalarchitecture of the system. We show that TOI-178 harbours at least six planets in the super-Earth to mini-Neptuneregimes with radii ranging from 1.177±0.074 to 2.91±0.11 Earth radii, and periods of 1.91, 3.24, 6.56, 9.96, 15.23, and20.71 d. All planets but the innermost one form a 2:4:6:9:12 chain of Laplace resonances, and the planetary densitiesshow important variations from planet to planet, jumping from 0.90+0.16

−0.21 to 0.15+0.03−0.04 times the Earth density between

planets c and d. Using Bayesian interior structure retrieval models, we show that the amount of gas in the planetsdoes not vary in a monotoneous way as one could expect from simple formation and evolution models, contrarily toother known systems in chain of Laplace resonances. The brightness of TOI-178 (H=8.76 mag, J=9.37 mag, V=11.95mag) allows for precise characterisation of its orbital architecture as well as of the physical nature of the six presentlyknown transiting planets it harbours. The peculiar orbital configuration and the diversity in average density amongthe planets in the system will enable the study of planetary interior structures and atmospheric evolution providingimportant clues on the formation of super-Earths and mini-Neptunes.

Key words. see on AA website. Transits · Mean-motion resonance · Laplace resonance · CHEOPS · TESS · NGTS ·SPECULOOS

Article number, page 1 of 31

A&A proofs: manuscript no. output


A. Leleu et al: Six transiting planets and a chain of Laplace resonances in TOI-178

1. Introduction

Since the discovery of the first exoplanet orbiting a Sun-likestar by Mayor & Queloz (1995), the diversity of observedplanetary systems has continued to challenge our under-standing of their formation and evolution. As an ongoingeffort to understand these physical processes, observationalfacilities strive to get the full picture of exoplanetary sys-tems by looking for additional candidates to known sys-tems, along with better constraining the orbital architec-ture, radii and masses of the known planets.

In particular, chains of planets in mean-motion reso-nances (MMRs) are ‘Rosetta Stones’ of the formation andevolution of planetary systems. Indeed, our current under-standing of planetary system formation theory implies thatsuch configurations are a common outcome of protoplane-tary discs: slow convergent migration of a pair of planets inquasi-circular orbits leads to a high probability of capturein first order MMRs - the period ratio of the two planetsis equal to (k + 1)/k, with k an integer (Lee & Peale 2002;Correia et al. 2018). As the disc strongly damps the ec-centricities of the protoplanets, this mechanism can repeatitself, trapping the planets in a chain of MMRs, leading tovery closely packed configurations (Lissauer et al. 2011).However, resonant configurations are not most common or-bital arrangements (Fabrycky et al. 2014). As the proto-planetary disc dissipates, the eccentricity damping lessens,which can lead to instabilities in packed systems (see for ex-ample Terquem & Papaloizou 2007; Pu & Wu 2015; Izidoroet al. 2017).

For planets that remained in resonance, and are closeenough to their host star, tides become the dominant forcethat affect the architecture of the systems, which can thenlead to a departure of the period ratio from the exact res-onance (Henrard & Lemaitre 1983; Papaloizou & Terquem2010; Delisle et al. 2012). In some near-resonant systems,such as HD158259, the tides seem to have pulled the con-figuration entirely out of resonance (Hara et al. 2020). How-ever, through gentle tidal evolution it is possible to retain aresonant state even with null eccentricities, through three-body resonances (Morbidelli 2002; Papaloizou 2015). Suchsystems are too fine-tuned to result from scattering events,and hence can be used to constrain the outcome of proto-planetary discs (Mills et al. 2016).

A Laplace resonance, in reference to the configurationof Io, Europa, and Ganymede, is a three body resonancewhere each consecutive pair of bodies are in, or close to,two planet MMRs. To date, very few systems were observedin a chain of Laplace resonances : Kepler-60 (Goździewskiet al. 2016), Kepler-80 (MacDonald et al. 2016), Kepler-223(Mills et al. 2016), and Trappist-1 (Gillon et al. 2017; Lugeret al. 2017). No radial velocity follow-up has been made forthese system so far, mainly due to the relative faintness oftheir host star in the visible (V magnitudes greater than∼14). However, TTVs could be used to estimate the massof their planets, see for example Agol et al. (2020) for thecase of Trappist-1.

In this study, we present photometric and RV observa-tions of TOI-178, a V = 11.95mag, K-type star, first ob-served by TESS in its sector 2. We jointly analyse the pho-tometric data of TESS, two nights of NGTS and SPECU-

? CHEOPS Fellow?? Winton Fellow

??? Juan Carlos Torres Fellow

LOOS data, and 285 hours of CHEOPS observations,along with forty-six ESPRESSO radial velocity points. Thisfollow-up effort allows us to decipher the architecture of thesystem and demonstrate the presence of a chain of Laplaceresonances between the five outer planets.

We begin in section 2 by presenting the rationale thatled to the CHEOPS observation sequence (two visits to-taling 11 d followed by two shorter visits). In section 3 wedescribe the parameters of the star. In section 4, we presentthe photometric and RV data we use in the paper. In sec-tion 5, we show how these data led us to the detection of6 planets, the five outer one being in a chain of Laplaceresonances, and to constrain their parameters. In section 6,we explain the resonant state of the system and discuss itsstability. In section 6.4, we describe the transit timing vari-ations (TTVs) that this system could potentially exhibit inthe coming years. Conclusions are presented in section 8.

2. CHEOPS observation strategy

CHEOPS observation consisted in one long double visit (11days) followed by to short visits (a few hours each) at pre-cise dates. We explain in this section how we came up tothis particular observation strategy. Details on all data usedand acquired, as well as on their analysis, are presented inthe following sections.

The first release of candidates from the TESS alerts ofSector 2 included three planet candidates in TOI-178 withperiods of 6.55 d, 10.35 d, and 9.96 d, respectively. Basedon these data, TOI-178 was identified as a potential co-orbital system (Leleu et al. 2019) with two planets oscillat-ing around the same period. This prompted ESPRESSO RVmeasurements and two sequences of simultaneous ground-based photometric observations with NGTS and SPECU-LOOS. From the latter, no transit was observed for TOI-178.02 (P = 10.35 d) in September 2019, however a transitascribed to this candidate was detected one month later byNGTS and SPECULOOS. The absence of transit of TOI-178.02 noted above, combined with the three transits ob-served by TESS at high SNR (above 10), was interpretedas an additional sign of the strong TTVs expected in a co-orbital configuration. This solution was supported by RVdata that were consistent with horseshoe orbits of similarmass objects (Leleu et al. 2015).

A continuous 11 d CHEOPS observation (split in twovisits for scheduling reasons) was therefore performed inAugust 2020 in order to confirm the orbital configurationof the system, as the instantaneous period of both mem-bers of the co-orbital pair would always be smaller than11 d, and thus at least one transit of both targets shouldbe detectable. Analysis of this light curve led to the con-firmation of the presence of two of the planets already dis-covered by TESS (in this study denoted as planets d and ewith periods of 6.55 d and 9.96 d, respectively) and the de-tection of two new inner transiting planets (denoted plan-ets b and c, respectively with periods of 1.9 d and 3.2 d).However, one of the planets belonging to the proposed co-orbital pair (period of 10.35 d) was not apparent in the lightcurve. A potential hypothesis was that the first and thirdtransit of the TOI-178.02 candidate (P = 10.35 d) duringTESS Sector 2 belonged to a planet of twice the period(20.7 d), while the second transit would belong to anotherplanet of unknown period. This scenario was supported bythe two ground-based observations mentioned above, as a



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Fig. 1: Light curves from TESS Sector 2 described in Sections 2 and 4.1.1. Unbinned data are shown as grey points,and data in 30 minutes bins are shown as black circles. The best fitting transit model for the system is shown in black,with the associated parameter values shown in Tables 3 and 4. The positions of the transits are marked with colouredlines according to the legend. The photometry before and after the mid-sector gap are shown in the top and bottompanels, respectively. As the first transit of planet f (thick teal line) occurred precisely between the two transits of thesimilarly-sized planet g (period of 20.71 d - red lines), the three transits were originally thought to have arisen due to asingle planet, which was originally designated TOI-178.02 with a period of 10.35 d (see section 2).

P = 20.7 d planet would not transit during the Septem-ber 2019 observation window. Using the ephemerides fromfitting the TESS, NGTS, and SPECULOOS data, we pre-dicted the mid-transit of the 20.7 days candidate to occurbetween UTC 14:06 and 14:23 on 2020-September-07 with98% certainty. A third visit of CHEOPS observed the sys-tem around this epoch for 13.36 h and confirmed the pres-ence of this new planet (g) at the predicted time, with con-sistent transit parameters. Further analysis of all availablephotometric data found two possibilities for the unknownperiod of the additional planet: ∼ 12.9 d or ∼ 15.24 d.

Careful analysis of the whole system additionally re-vealed that planets c, d, e and g were in a Laplace reso-nance. In order to fit in the resonant chain, the unknownperiod of the additional planet could have only two values:P = 13.4527 d, or P = 15.2318 d, the latter value being alsomore consistent with the RV data. A fourth CHEOPS visitwas therefore scheduled on Oct. 03 and detected the tran-sit for the additional planet at a period of 15.23186+0.00011

−0.00012day.

As we will detail in the next sections, using the newobservations presented in this paper we can confirm thedetection of a 6.56 and 9.96 d period planets by TESS, andannounce the detection of planets at 1.91, 3.24, 15.23, and

20.71 d (see Tables 3 and 4 for all the parameters of thesystem).

3. TOI-178 Stellar Characterisation

Forty-six ESPRESSO observations (see Sect. 4.2) of TOI-178, a V = 11.95mag K-dwarf, have been used to deter-mine the stellar spectral parameters. These observationshave first been shifted and stacked to produce a combinedspectrum. We have then used the publicly available spectralanalysis package SME (Spectroscopy Made Easy; Valenti& Piskunov 1996; Piskunov & Valenti 2017) version 5.22to model the co-added ESPRESSO spectrum. We selectedthe ATLAS12 model atmospheres grids (Kurucz 2013) andatomic line data from VALD to compute synthetic spectrawhich was fitted to the observations using a χ2-minimisingprocedure. We modelled different spectral lines to obtaindifferent photospheric parameters and started with the linewings of Hα which are particularly sensitive to the stellar ef-fective temperature Teff . We then proceeded with the metalabundances and the projected rotational velocity v sin i?which were modelled with narrow lines between 5900 and6500 Å. We found similar values for [Fe/H], [Ca/H], and[Na/H]. The macro- and micro turbulent velocities werefixed to 1.2 km s−1 and 1.0 km s−1, respectively, and mod-


http://www.stsci.edu/~valenti/sme.html

http://vald.astro.uu.se


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Fig. 2: Light curves from simultaneous observations of TOI-178 by NGTS (green stars) and SPECULOOS-South (pur-ple triangles), described in sections 4.1.3 and 4.1.4, respec-tively. Unbinned data are shown as grey points, and data in15 minute bins are shown as green stars (NGTS) and pur-ple triangles (SPECULOOS). The observations occurred on2019-September-11 (top panel) and 2019-October-12 (bot-tom panel). The transit model is shown in black. The po-sition of the odd transit of candidate TOI-178.02 is shownwith a dashed red line, the transit of planet g (which corre-sponds to an even transit of TOI-178.02) is shown with thesolid red line, and the transits of planet b are shown withpurple lines.

elled with a radial-tangential profile. The surface gravitylog g was constrained from the line wings of the Ca i triplet(6102, 6122, & 6162Å) and the Ca i 6439 Å line with a fixedTeff and Ca abundance.

We checked our model with the Na i doublet sensitive toboth Teff and log g, and finally we also tested the MARCS2012 (Gustafsson et al. 2008) model atmosphere grids. Thederived and finally adopted parameters are listed in Ta-ble 1. The SME results are in agreement with the empiricalSpecMatch-Emp (Yee et al. 2017) code which fits the stellaroptical spectra to a spectral library of 404 M5 to F1 starsresulting in Teff = 4316 ± 70 K, log g = 4.45 ± 0.15, and[Fe/H] = −0.29 ± 0.05 dex. We also used ARES+MOOG(Sousa 2014; Sousa et al. 2015; Sneden 1973) to do the spec-troscopic analysis on the same combined ESPRESSO spec-tra and although we derived consistent parameters (Teff =4499±227 K, log g = 4.38±0.62, and [Fe/H] = −0.34±0.1),the large errors are indicative of the difficulties that equiv-alent width methods have in colder stars as this one wheremore spectral lines are more crowded in the spectrum.

Using these precise spectral parameters as priors onstellar atmospheric model selection, we determined the ra-dius of TOI-178 using the infrared flux method (IRFM;Blackwell & Shallis 1977) in a Markov-chain Monte Carlo(MCMC) approach. The IRFM computes the stellar angu-lar diameter and effective temperature via comparison be-tween observed broadband optical and infrared fluxes andsynthetic photometry obtained from convolution of the con-sidered filter throughputs, using the known zero-point mag-nitudes, with the stellar atmospheric model, with the stel-

Table 1: Stellar properties of TOI-178 including the meth-ods used to derive them.

TOI-1782MASS J00291228-3027133Gaia DR2 2318295979126499200TIC 251848941TYC 6991-00475-1Parameter Value Noteα [J2000] 00h29m12.30s 1δ [J2000] -30◦27

′13.46

′′1

µα [mas/yr] 149.95±0.07 1µδ [mas/yr] -87.25±0.04 1$ [mas] 15.92±0.05 1RV [km s−1] 57.4±0.5 1V [mag] 11.95 2G [mag] 11.15 1J [mag] 9.37 3H [mag] 8.76 3K [mag] 8.66 3W1 [mag] 8.57 4W2 [mag] 8.64 4Teff [K] 4316±70 spectroscopylog g [cgs] 4.45±0.15 spectroscopy[Fe/H] [dex] -0.23±0.05 spectroscopyv sin i? [km s−1] 1.5±0.3 spectroscopyR? [R�] 0.651±0.011 IRFMM? [M�] 0.647+0.035

−0.032 isochronest? [Gyr] 7.1+6.2

−5.4 isochronesL? [L�] 0.132±0.010 from R? and Teff

ρ? [ρ�] 2.35±0.17 from R? and M?

Notes. [1] Gaia Collaboration et al. (2018), [2] Høg et al. (2000),[3] Skrutskie et al. (2006), [4] Wright et al. (2010)

lar radius then calculated using the parallax of the star.For this study, we retrieved the Gaia G, GBP, and GRP,2MASS J, H, and K, and WISE W1 and W2 fluxes andrelative uncertainties from the most recent data releases(Skrutskie et al. 2006; Wright et al. 2010; Gaia Collab-oration et al. 2018, respectively), and utilised the stellaratmospheric models from the atlas Catalogues (Castelli& Kurucz 2003), to obtain R? = 0.651 ± 0.011R�, andTeff = 4352 ± 52 K, in agreement with the spectroscopicTeff used as a prior.

We inferred the mass and age of TOI-178 using stel-lar evolutionary models, using as inputs Teff, R? andFe/H, with evolutionary tracks and isochrones generated bytwo grids of models separately; the PARSEC1 v1.2S code(Marigo et al. 2017) and the CLES code (Code Liégeoisd’Évolution Stellaire; Scuflaire et al. 2008), with the re-ported values representing a careful combination of resultsfrom both sets of models. This was done as the sets of mod-els differ slightly in approaches (reaction rates, opacity andovershoot treatment, and helium-to-metal enrichment ra-tio), and thus by comparing masses and ages derived fromboth grids it is possible to include systematic uncertaintieswithin modelling the position of TOI-178 on evolutionarytracks and isochrones. A detailed discussion of combiningthe PARSEC and CLES models to determine masses and

1 Padova and Trieste Stellar Evolutionary Codehttp://stev.oapd.inaf.it/cgi-bin/cmd.



ages can be found in Bonfanti et al. (submitted). For thisstudy we deriveM? = 0.647+0.035

−0.032M� and t? = 7.1+6.2−5.4 Gyr.

All stellar parameters are shown in Table 1.

4. Data

4.1. Photometric data

In order to determine the orbital configuration of the TOI-178 planetary system we obtained photometric time-seriesobservations from multiple telescopes, as detailed below.

4.1.1. TESS

Listed as TIC 251848941 in the TESS Input Catalog(TIC; Stassun et al. 2018, 2019), TOI-178 was observed byTESS in Sector 2, camera 2 from 2018-August-22 to 2018-September-20. The individual frames were processed into2minute cadence observations and reduced by the ScienceProcessing Operations Center (SPOC; Jenkins et al. 2016)into light curves made publicly available at the MikulskiArchive for Space Telescopes (MAST). For our analysis weretrieved the Presearch Data Conditioning Single AperturePhotometry (PDCSAP) light curve data, using the defaultquality bitmask, that has undergone known systematic cor-rection (Smith et al. 2012; Stumpe et al. 2014). Lastly, werejected data points flagged as of bad quality by the SPOC(QUALITY > 0) and those with Not-a-Number flux or fluxerror values. After these quality cuts, the TESS light curveof TOI-178 contained 18,316 data points spanning 25.95 d.The full dataset with the transits of the six identified plan-ets is shown in Fig. 1.

4.1.2. CHEOPS

CHEOPS, the first ESA small-class mission, is dedicatedto the observation of bright stars (V . 12 mag) that areknown to host planets and performs ultra high-precisionphotometry, with the precision being limited by stellar pho-ton noise of 150 ppm/min for a V =9 magnitude star. TheCHEOPS instrument is composed of a f/8 Ritchey-Chretienon-axis telescope (∼30 cm diameter) equipped with a sin-gle frame-transfer back-side illuminated CCD detector. Thesatellite was successfully launched from Kourou (FrenchGuiana) into a ∼700 km altitude Sun-synchronous orbiton December 18th 2019. CHEOPS took its first image onFebruary 7th 2020 and, after passing the In-Orbit Commis-sioning (IOC) phase, routine operations started on March25th 2020. More details on the mission can be found inBenz et al. (2020), and the first results have recently beenpresented in Lendl et al. (2020).

The versatility of the CHEOPS mission allows for space-based follow-up photometry of planetary systems identifiedby the TESS mission. This is particularly useful to com-plete the inventory of multi-planetary systems whose outertransiting planets have periods beyond ∼10 days.

We obtained four observation runs (or visits) of TOI-178with CHEOPS between 2020-August-04 and 2020-October-03 as part of the Guaranteed Time, totalling 11.88 dayson target with the observing log shown in Table 2. Themajority of this time was spent during the first two vis-its (with lengths of 99.78 h and 164.06 h respectively) thatwere conducted sequentially from 2020-August-04 to 2020-August-15 so to achieve a near continuous 11 day photo-

Fig. 3: Extraction of 80×80 arcsec of the CHEOPS Field-of-View for two different data frames at the beginning (top)and the end (bottom) of the second visit (see Table 2). TheTOI-178 PSF is shown at the center with the DEFAULTDRP photometric aperture represented by the dashed blackcircles. The telegraphic pixel location which appeared closeto the end of the observation is marked by the red circle.

metric time series. The runs were split due to schedulingconstraints with 0.84 h gap between visits. The third andfourth visits were conducted to confirm the additional plan-ets predicted in the scenario presented in Section 2. Theytook place on 2020-September-07 and 2020-October-03, andlasted for 13.36 h and 8.00 h, respectively.

Due to the low-Earth orbit of CHEOPS, the spacecraft-target line of sight was interrupted by Earth occulta-tions, and along with passages through the South At-lantic Anomaly (SAA) where no data were downlinked.This resulted in gaps in the photometry on CHEOPS orbittimescales (around 100 min). For our observations of TOI-178, this resulted in light curve efficiencies of 51%, 54%,65%, and 86%. For all four visits we used an exposure timeof 60 s.

These data were automatically processed with the lat-est version of the CHEOPS data reduction pipeline (DRP



v12; Hoyer et al. 2020). This includes performing imagecalibration (bias, gain, non-linearity, dark current, and flatfield), and conducting instrumental and environmental cor-rections (cosmic rays, smearing trails of field stars and back-ground). It also performs aperture photometry for a set ofsize-fixed apertures: R=22.5′′(RINF), 25′′(DEFAULT) and30′′(RSUP), plus one extra aperture which minimizes thecontamination from nearby field stars (ROPT) which in thecase of TOI-178 was set in R=28.5′′. The DRP estimatesthe level of contamination by simulating the Field-of-Viewof TOI-178 using GAIA star catalogue (Gaia Collabora-tion et al. 2018) to determinate the location and flux of thestars through the duration of the visit. In the case of TOI-178, the mean contamination level was below the 0.1% andmodulated mostly by the rotation around the target of anearby star of Gaia G=13.3mag at a projected sky distanceof 60.8′′from the target. The contamination as a functionof time (or equivalently as a function of roll angle of thesatellite) is provided as a product of the DRP for furtherdetrending (see Sect.5.1). For the second visit, careful re-moval of one telegraphic pixel (a pixel with an non-stableabnormal behaviour during the visit) within the photomet-ric aperture was needed. The location of this telegraphicpixel is shown in red in Fig.3, and details on the detectionand correction are described in Appendix A. Following thereductions, we found that the light curves obtained usingthe DEFAULT aperture (R=25′′) yielded the lowest RMSfor all visits and so are used for this study.

Lastly, it has become apparent that due to the nature ofthe CHEOPS orbit and rotation of the CHEOPS field, pho-tometric, non-astrophysical short-term trends either from avarying background, nearby contaminating source or othersources, can be found in the data on orbital timescales.These systematics can be successfully removed via a lin-ear detrending method against basis vectors of concern(e.g. Lendl et al. 2020, Bonfanti et al. submitted, Delrezet al. submitted). In this study we carefully inspected thelight curves, and determined that by decorrelating the lightcurve against background and contamination estimates,and against CHEOPS roll angle, any remaining systemat-ics were removed. For the final run, the background washeavily influenced by scattered light from the Moon andthus, this glint contribution was fitted by a N = 40 splineas an additional basis vector within the detrending and re-moved. Following this, the average noise over a 3 h slidingwindow for the four visits of the G = 11.15mag target wasfound to be 63.9, 64.2, 66.3, and 75.8 ppm respectively. Inall cases, this marginally improved upon the precision ofthe light curves that we had previously simulated for theseobservation windows using the CHEOPSim tool (Futyan et al.2020).

4.1.3. NGTS

The Next Generation Transit Survey (NGTS; Wheatleyet al. 2018) facility consists of twelve 20 cm diameter robotictelescopes and is situated at the ESO Paranal Observatoryin Chile. The individual NGTS telescopes have a wide field-of-view of 8 square-degrees and a plate scale of 5 ′′ pixel−1.The DONUTS auto-guiding algorithm (McCormac et al.2013) affords the NGTS telescopes sub-pixel level guiding.Simultaneous observations using multiple NGTS telescopeshave been shown to yield ultra-high precision light curvesof exoplanet transits (Bryant et al. 2020; Smith et al. 2020).

TOI-178 was observed using NGTS multi-telescope ob-servations on two nights. On UT 2019 September 10 TOI-178 was observed using six NGTS telescopes during thepredicted transit event of the TESS candidate TOI-178.02.However, the NGTS data for this night rule out a tran-sit occurring during the observations. A second predictedtransit event of TOI-178.02 was observed on the night UT2019 October 11 using seven NGTS telescopes, and on thisnight the transit event was robustly detected by NGTS.A total of 13,991 images were obtained on the first night,and 12,854 were obtained on the second. For both nightsthe images were taken using the custom NGTS filter (520- 890 nm) with an exposure time of 10 s. All NGTS obser-vations of TOI-178 were performed with airmass < 2 andunder photometric sky conditions.

The NGTS images were reduced using a custom pipelinewhich uses the SEP library to perform source extractionand aperture photometry (Bertin & Arnouts 1996; Bar-bary 2016). A selection of comparison stars with brightness,colour, and CCD position similar to TOI-178 were iden-tified using GAIA DR2 (Gaia Collaboration et al. 2018).More details on the photometry pipeline are provided inBryant et al. (2020).

The NGTS light curves are presented in Fig. 2, andshow transit events for planet b and planet g on the nightsof 2019 September 11 and 2019 October 1 respectively.

4.1.4. SPECULOOS

The SPECULOOS Southern Observatory (SSO; Gillon2018; Burdanov et al. 2018; Delrez et al. 2018) is locatedat ESO’s Paranal Observatory in Chile and is part of theSPECULOOS project. The facility consists of a network offour robotic 1-m telescopes (Callisto, Europa, Ganymede,and Io). Each SSO robotic telescope has a primary aper-ture of 1m and a focal length of 8m, and is equippedwith a 2k×2k deep-depletion CCD camera whose 13.5 µmpixel size corresponds to 0.35" on the sky (field of view =12′x12′). Observations were performed on the nights start-ing the 10th of October 2019 (for ' 8 hours) and the 11thof November 2019 (for ' 8 hours) with three SPECULOOStelescopes on sky simultaneously (SSO/Io, SSO/Europa,and SSO/Ganymede). These observations were carried outin a Sloan i’ filter with exposure times of 10 s. A smalldefocus was applied to avoid saturation as the target wastoo bright for SSO. Light curves were extracted using theSSO pipeline (Murray et al. 2020) and shown in purpleon Fig.2. For each observing night, the SSO pipeline usesthe casutools software (Irwin et al. 2004) to perform auto-mated differential photometry and correct from systematicscaused by time-varying telluric water vapor.

4.2. ESPRESSO Data

The radial velocity data we analyse consists in 46ESPRESSO points2. Each measurement is taken in HRmode with an integration time of 20 min in single UT modeand slow read-out (HR 21). The source on fiber B is theFabry-Perot interferometer. Observations were made witha maximum airmass of 1.8 and minimum 30◦ separationwith the Moon.2 The first 32 ones come from program 0104.C-0873(A), the last14 from guaranteed time observations.


https://www.speculoos.uliege.be/cms/c_4259452/en/speculoos


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Fig. 4: Similar to Fig. 1, but instead displaying data from the four CHEOPS visits described in 4.1.2. Top panel: the 11 dobservation. Transits of planets c and d at ∼ 2459075 BJD occur too close for their corresponding lines to be individuallyvisible in the figure. Bottom left panel: subsequent observation scheduled to confirm the presence of a planet with a 20.7d period (planet g), which overlaps with a transit of planet e. Bottom right panel: final observation scheduled to confirmthe presence of a planet fitting in the Laplace resonance (planet f, with a period of ∼15.23 d period), which overlapswith a transit of planet b.

Table 2: The log of CHEOPS observations of TOI-178.

visit Identified Start date Duration Data points File key Efficiency Exp. Time# planets [UTC] [h] [#] [%] [s]1 b,c,d,e 2020-08-04T22:11:39 99.78 3030 CH_PR100031_TG030201_V0100 51 602 b,c,d 2020-08-09T02:48:39 164.04 5280 CH_PR100031_TG030301_V0100 54 603 e,g 2020-09-07T08:06:44 13.36 521 CH_PR100031_TG030701_V0100 65 604 b,f 2020-10-03T18:51:46 8.00 413 CH_PR100031_TG033301_V0100 86 60

The measurements span from 29 Sep. 2019 to 20 Jan.2020 and have an average nominal error bar of 93 cm/s.We also include in our analysis the time-series of Hα mea-surements, the full width half maximum (FWHM) and theS-index. The velocity and ancillary indicators are extractedfrom the spectra with the standard ESPRESSO pipeline v2.0.0 (Pepe et al. 2020). The RV time-series with nominalerror bars is shown in Fig. 5.

5. Detections and parameters estimations

We analyse the photometric and spectral data separatelyand jointly, which we describe in the following sections. This

approach allows us to independently detect the planets inthe system, and subsequently verify them in a global fit.

5.1. Analysis of the photometry

5.1.1. Identification of the solution

The first release of candidates from the TESS alerts of Sec-tor 2 included three planet candidates in TOI-178 with peri-ods of 6.55 d, 10.35 d, and 9.96 d, which transited 4, 3, and 2times, respectively. In addition, our analysis of this datasetwith the DST (Cabrera et al. 2012) yielded two additionalcandidates, a clear 3.23 d signal, and a fainter 1.91 d one.Upon receiving the visits 1 and 2 from CHEOPS (Table 2),



58760 58780 58800 58820 58840 58860Time (BJD - 2400000)

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TOI 178 ESPRESSO radial velocities

Fig. 5: ESPRESSO RV data of TOI-178.

a study of the CHEOPS data alone with successive appli-cations of the BLS algorithm (Kovács et al. 2002) retrievedthe 6.55 d 3.23 d, and 1.91 d signal in phase agreement withthe TESS data. An additional dim, consistent in epoch witha transit of the 9.9d candidate was also identified, but couldalso have marginally correspond to a transit of the 10.35 dcandidate.

Taking TESS sector 2, NGTS visits of September andOctober 2019, and CHEOPS visits 1 and 2, we individuallypre-detrended the datasets and subtracted the signal of the6.55 d 3.23 d, and 1.91 d candidates. We then applied theBLS algorithm a first time on this dataset, saved the mostlikely candidate of period P1, and removed the correspond-ing signal, and applied the BLS a second time to obtain asecond candidate of period P2. That created a first pair ofcandidates c0,0. We then repeated this process, but ignor-ing the result of the BLS for periods less that 0.2 d awayfrom P1 for the first candidate of the pair, while taking thehighest peak on the second iteration of the BLS, leading tothe pair c1,0. We repeated this process 25 times, yieldingto 25 potential pairs of candidates c0≤i≤4,0≤j≤4, where iand j are the number of peaks that have been ignored inthe first or second iteration of the BLS, respectively. Forall of these potential solutions we modeled the transits ofthe 5 candidates : 1.91 d, 3.23 d, 6.55 d, and ci,j ; using thebatman package (Kreidberg 2015) and ran an MCMC on thepre-detrended lightcurve to estimate the relative likelihoodof the different ci,j . The 1.91 d, 3.23 d, 6.55 d, 9.96 d, and20.71 d model was favoured, and explained all the signifi-cant dims observed in the NGTS/SPECULOOS data (Fig.2) and the first visit of CHEOPS (Fig. 4 - top). This so-lution was later confirmed by the predicted double transitobserved by the 3rd visit of CHEOPS (Fig. 4 - bottom left).

Applying the BLS algorithm on the residuals of theavailable photometric data, Two mutually exclusive peaksappeared: ∼ 12.9 d and ∼ 15.24 d, the former being slightlyfavoured by the BLS analysis. However, the global fit of thelightcurve favoured the ∼ 15.24 d signal. In addition, thissolution was very close to the period that would fit the res-onant structure of the system, see section 6. The ∼ 15.24 dcandidate was confirmed by a 4th CHEOPS visit (Fig. 4 -bottom right). In the next section we develop the charac-terisation of this 6-planets solution: 1.91 d, 3.23 d, 6.55 d,9.96 d, ∼ 15.24 and 20.71 d.

5.1.2. Determination of radii and orbital parameters

To detect orbiting bodies, and characterise the system weperformed a fit of the TESS, NGTS, and CHEOPS pho-tometry. As the NGTS and SPECULOOS data presented

in Fig. 2 cover the same epoch of observations, we only in-cluded the NGTS data in our fit as it had a smaller RMSphotometric scatter.

The fit was performed using the allesfitter package(Günther & Daylan 2020). Wide, uniform priors for eachplanet were placed on the transit parameters Rp/R?, (R?+Rp)/a, cos i, T0, and P ; where Rp and R? are respectivelythe radii of the planet and star, a is the semi-major axis, i isthe orbital inclination, T0 is the time of inferior conjunction,and P is the period. Gaussian priors with σ = 0.1 wereplaced on the quadratic limb darkening parameters q1 andq2 (the parameterisation given by Kipping 2013) for eachinstrument, with the mean values calculated according toClaret & Bloemen (2011). We also placed a uniform prior onρ?, calculated as described in section 3. The eccentricity andargument of periastron components for each planet used inthe fitting was fixed to 0, as justified in section 6.3.

Systematic trends were removed from the CHEOPSlight curves before their inclusion in the global fit using thepycheops package (Maxted et al., in prep.). Each visit wasdetrended using a set of linear basis vectors that optimisedthe Bayesian Evidence Z (chosen from background, con-tamination, linear and quadratic functions of detection xyposition, and trigonometric functions of roll angle). Due tothe small angular separation of the moon for the final visit,we also detrending with a linear basis vector constructedfrom a spline function fit fitting the flux as a function ofroll angle, centred on the direction of the moon.

The time correlated trends in the TESS light curve weremodelled simultaneously with the transit parameters us-ing a Gaussian process with a Matérn 3/2 kernel which isimplemented in allesfitter using the celerite package(Foreman-Mackey et al. 2017a). The natural logarithms ofthe hyperparameters were allowed to vary between -5 and5 for log(σ) and 5 and 15 for log(ρ).

The best-fitting values and uncertainties associated witheach parameter were calculated using dynamic nested sam-pling, which is implemented in allesfitter using thedynesty package. The best-fitting transit parameters as-sociated with each of the planet are displayed in Table 4.1.

The results from the model are displayed in Tables 3and 4.

This analysis yields a solution of the system that is bestexplained by 6 signals of planetary nature:

– a 275 ppm signal of 1.65 h duration, with a time ofconjunction of 2458741.6371 BJD, and a period of1.914557 d.


– a 1383 ppm signal of 2.280 h duration, with a timeof conjunction of 2458747.1465 BJD, and a period of6.557694 d.






Table 3: Fitted and derived results for planets b, c, and d associated with the fits to the photometry and spectroscopydescribed in sections 5.1 and 5.2, respectively.

Parameter (unit) b c dFitted parameters (photometry)

Rp/R? 0.01658± 0.00098 0.0241+0.0012−0.0011 0.03719± 0.00074

(R? +Rp)/a 0.1181+0.0028−0.0026 0.0836+0.0018

−0.0017 0.0530+0.0014−0.0012

cos i 0.035± 0.021 0.034+0.013−0.017 0.0272± 0.0027

t0 (BJD-TBD) 2458741.6371+0.0037−0.0030 2458741.4790± 0.0025 2458747.14645+0.00071

−0.00075

P (d) 1.914557+0.000016−0.000018 3.238458+0.000020

−0.000018 6.557694± 0.000013Fitted parameters (spectroscopy)

K (ms−1) 1.05+0.25−0.30 2.77+0.22

−0.33 1.34+0.31−0.39

Derived parametersδtr (ppm) 275+32

−31 580+60−52 1383+55

−54

R?/a 0.1162+0.0028−0.0025 0.0816+0.0018

−0.0017 0.0511+0.0014−0.0012

a/R? 8.61± 0.20 12.25± 0.27 19.58+0.46−0.51

Rp/a 0.00193+0.00012−0.00011 0.001965+0.00012

−0.000097 0.001899+0.000074−0.000066

Rp (R⊕) 1.177± 0.074 1.710+0.094−0.082 2.640± 0.069

a (AU) 0.02604± 0.00075 0.0371± 0.0010 0.0592± 0.0018i (deg) 88.0± 1.2 88.04+0.97

−0.74 88.44± 0.16b 0.30± 0.18 0.42+0.15

−0.21 0.532+0.041−0.043

t14 (h) 1.649+0.064−0.11 1.89+0.13

−0.17 2.054± 0.043t23 (h) 1.590+0.067

−0.12 1.78+0.14−0.18 2.054± 0.043

Teq (K) 952± 19 798± 16 631± 13Mp (M⊕) 1.51+0.38

−0.45 4.76+0.55−0.67 2.90+0.72

−0.92

ρp (ρ⊕) 0.91+0.26−0.33 0.90+0.16

−0.21 0.15+0.03−0.04

We denote this series of planetary signals respectively asTOI-178 b, c, d, e, f, and g.

The signals were detected with SNRs of 8.47, 9.72,25.23, 15.61, 15.2, and 14.52, respectively, with the num-ber of transits for each planetary signal being: 19, 11, 6, 4,3, and 4. When incorporating the stellar radius calculatedas described in section 3, the planets have radii of 1.177,1.710, 2.640, 2.169, 2.379, and 2.91 R⊕, with the inner twoplanets falling either side of the radius valley (Fulton et al.2017).

The transits of all detected planets are shown in Figs. 1and 4, with the phase folded transits in the CHEOPS datapresented in Fig. 6.

5.2. Analysis of the radial velocities

5.2.1. Detections

In this section, we consider the forty-six ESPRESSO datapoints only, and look for potential planet detections. Ouranalysis follows the same steps as Hara et al. (2020), and isdescribed in detail in Appendix B. To search for potentialperiodicities, we computed the `1 periodogram3 of the RV,as defined in Hara et al. (2017). This method outputs afigure which has a similar aspect as a regular periodogram,but with fewer peaks due to aliasing. The peaks can beassigned a false alarm probability (FAP), whose interpreta-tion is close to the FAP of a regular periodogram peak.

A preliminary analysis of ancillary indicators Hα,FWHM, bisector span (Queloz et al. 2001) andlogR′HK (Noyes 1984) revealed that they exhibit statisti-cally significant periodicities at ≈ 36 days and ≈ 16 days,such that stellar activity effects are to be expected in the3 https://github.com/nathanchara/l1periodogram

RVs, especially at these periods. In our analysis, stellar ac-tivity has been taken into account both with a linear modelconstructed with activity indicators smoothed with a Gaus-sian process regression similarly to Haywood et al. (2014),which we call the base model, and with a Gaussian noisemodel with a white, correlated (Gaussian kernel) and quasiperiodic component.

Radial velocity signals found to be statistically signif-icant might vary from one activity model to another. Toannounce robust detections, we tested whether signal de-tections can be claimed for a variety of noise models, fol-lowing Hara et al. (2020). This approach consists in threesteps. First, we computed `1-periodogram of the data on agrid of models. The linear base models considered includean offset and smoothed ancillary indicators (Hα, FWHM,none or both), where the smoothing is done with a Gaussianprocess regression with a Gaussian kernel. The noise mod-els considered are Gaussian with three components: white,red with Gaussian kernel, and quasi-periodic. According tothe analysis of ancillary indicators, the quasi-periodic termof the noise is fixed at 36.5 days. We consider a grid of val-ues for each noise component (amplitude, decay time-scale),and compute the `1-periodograms. Secondly, we rank thenoise models with a procedure based on cross-validation.Finally, we examine the distribution of FAPs of each signalin the 20% highest ranked models

We here succinctly describe the conclusions of our anal-ysis, and refer the reader to Appendix B for a more de-tailed presentation. The `1-periodogram corresponding tothe highest ranked model is represented in Fig. 7. The peri-ods at which the peaks occur are shown in red, and accountfor most of the signals that might appear with varying as-sumptions on the noise/activity model. More precisely, wefind the following results. Note that these ones stem from



Table 4: Fitted and derived results for planets e, f, and g associated with the fits to the photometry and spectroscopydescribed in sections 5.1 and 5.2, respectively.

Parameter (unit) e f gFitted parameters (photometry)

Rp/R? 0.03057+0.00091−0.00098 0.0335± 0.0011 0.0410± 0.0014

(R? +Rp)/ap 0.03989+0.00098−0.00091 0.03001+0.00070

−0.00065 0.02462± 0.00056cos i 0.0233+0.0020

−0.0025 0.02214± 0.00100 0.02036+0.00068−0.00064

T0 (BJD-TBD) 2458751.4661+0.0014−0.0019 2458745.7177+0.0021

−0.0025 2458748.0293+0.0014−0.0012

P (d) 9.961872+0.000034−0.000038 15.231930+0.00011

−0.000085 20.709460+0.000077−0.000069

Fitted parameters (spectroscopy)K (ms−1) 1.62+0.41

−0.34 2.76+0.46−0.42 1.30+0.38

−0.59

Derived parametersδtr (ppm) 934+56

−60 1123+70−71 1680+110

−120

R?/a 0.03871+0.00094−0.00088 0.02904+0.00068

−0.00063 0.02365± 0.00053a/R? 25.83± 0.61 34.44± 0.79 42.28± 0.93Rp/a 0.001183± 0.000052 0.000973+0.000041

−0.000039 0.000969± 0.000043Rp (R⊕) 2.169± 0.079 2.379± 0.086 2.91± 0.11a (AU) 0.0782± 0.0023 0.1042± 0.0029 0.1280± 0.0036i (deg) 88.67+0.14

−0.12 88.731± 0.057 88.833+0.037−0.039

b 0.604+0.041−0.060 0.763+0.020

−0.022 0.861+0.012−0.014

t14 (h) 2.455+0.12−0.068 2.356+0.063

−0.059 2.189+0.060−0.055

t23 (h) 2.226+0.13−0.079 2.005+0.079

−0.075 1.579± 0.092Teq (K) 549± 11 475.8± 9.6 429.4± 8.5Mp (M⊕) 4.04+1.07

−0.95 7.94+1.45−1.52 4.14+1.26

−1.82

ρp (ρ⊕) 0.39+0.12−0.10 0.58+0.13

−0.12 0.19+0.05−0.09

an analysis of over 1300 noise models, and are not all ev-ident from Fig. 7, obtained with the highest ranked noisemodel.

– Our analysis yields consistent, significant detection ofsignals close to 3.2, ≈ 36 and ≈ 16 days. RV then allowsan independent detection of planet c. Signals at ≈ 36and ≈ 16 d appear in ancillary indicators, such that weattribute these apparent periodicities in RV to stellaractivity. The 16 days signal is, however, very likely to bepartly due to planet f. Indeed, when the activity signalsclose to 40 and 16 days are modelled and the signal oftransiting planets is removed, one finds a residual signalat 15.1 or 15.2 days, even though 16 and 15.2 days arevery close.

– We find signals, though not statistically significant, at6.5 and 9.8 days (consistent with planets d and e) and2.08 days, which is the one day alias (see Dawson &Fabrycky 2010) of 1.91 days (planet b).

– We do not consistently find a candidate near 20.7 days.However, a 20.6 days signal appears in the highestranked model, and stellar activity signal might hide thesignal corresponding to TOI-178 g.

– The signal at 43.3 days appearing in Fig. 7 might bea residual effect of an imperfect correction of the ac-tivity. It is however not strictly excluded that it mightstem from a planetary companion at 45 days. As will bediscussed in the conclusion, this period corresponds toone of the possibilities in order to continue the Laplaceresonance beyond planet g.

– Depending on the assumptions, hints at 1.2 or 5.6 days(aliases of each other) can appear.

For comparison, we performed the RV analysis with an it-erative periodogram approach. This one is able to show

signals corresponding to 15.2, 3.2 and 6.5 days, however,this one is unable to clearly establish the significance of the3.2 d signal, and fails to unveil candidates at 1.91 and 9.9days.

The photometric data allows to independently detect sixplanets at 1.91, 3.24, 6.56, 9.96, 15.23 and 20.71 days.As de-tailed in Appendix B, wee find that the phases of RV andphotometric signals are consistent within 2σ. We phase-fold the RV data at the periods given in Tables 3 and Ta-bles 4, which are shown in Fig. 8 with increasing periodsfrom top to bottom. The variations at 3.24 and 15.2 days,corresponding to the planets with the most significant RVsignals, are the clearest. As a final remark, the signals cor-responding to the transiting planets have been fitted, thestrongest periodic signals occur at 38 and 16.3 days, whichis compatible with the activity periods seen in the ancillaryindicators.

5.2.2. Mass and density estimations

To estimate the planetary masses, we fit circular orbits tothe radial velocities. As shown in section 6, for the systemto be stable, eccentricities cannot be greater than a fewpercents. We set as priors on Tc and period the posteriordistributions obtained from the fits to photometric datafrom Tables 3 and 4). This approach, as opposed to a jointfit, is justified by the fact that, here, RVs brings very littleinformation to the parameters constrained by photometryand vice-versa. Activity signals clearly are present in the RVdata, and depending on the activity model used, mass esti-mates may vary. To assess the model dependency of massestimates, we use two different activity models. Both in-clude as a linear predictor the smoothed Hα time series,as described above. In the first model, we represent activ-



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Fig. 6: Detrended CHEOPS light curves phase folded tothe periods of each of the planets, with signals of the otherplanets removed. Unbinned data are shown in grey points,data in bins of 15 min are shown in coloured circles, andsamples drawn from the posterior distribution of the globalfit are shown in coloured lines.

ity as a sum of two sine functions at 36 and 16 days anda correlated noise with an exponential kernel. This is mo-tivated by the fact that both periodicities appear the an-cillary indicators, but with different phases. The correlatednoise models low frequency variations, which also can beanticipated from the analysis of the indicators. The secondmodel of activity consists of a correlated Gaussian noisewith a quasi-periodic kernel. This analysis is presented indetail in Appendix B.5.

The mass estimation for each model are given in ta-bles B.2, 3, and 4, respectively. The 1σ intervals obtainedwith the two methods all have a large overlap. In Tables3 and 4, we give the mass and density intervals in a con-servative manner. The lower and upper bounds are respec-tively taken as the minimum lower bound and maximumupper bound obtained with the two estimation methods.The mass estimates are given as the mean of the estimatesobtained with the two methods, which are posterior medi-ans. We take this approach and do not select the error barsof one model or another, since model comparisons heavilydepend on prior chosen, which in our case would be rather

101 102

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1 periodogramPeak periods (d)

TOI 178, ESPRESSO RVs 1 periodogram

Fig. 7: `1 periodogram corresponding to the best noise mod-els in terms of cross-validation score, computed on a gridof frequencies from 0 to 0.55 cycles per day.

Table 5: Instantaneous distance from resonances for the 6planets of TOI-178, expressed in terms of near resonant an-gles φ. λ indicates the mean longitude of the planet specifiedin the subscript.

φi dφi/dt [deg/day] super period [day]

φ0 = 3λb − 5λc 8.2811+0.0065−0.0062 43.472+0.033

−0.034

φ1 = 1λc − 2λd 1.36886+7.8e−04−7.6e−04 262.99+0.15

−0.15

φ2 = 2λd − 3λe 1.38188+9.4e−04−9.7e−04 260.52+0.18

−0.18

φ3 = 2λe − 3λf 1.37131+8.0e−04−7.8e−04 262.52+0.15

−0.15

φ4 = 3λf − 4λg 1.37050+6.0e−04−6.0e−04 262.68+0.12

−0.11

arbitrary. We find planet masses and density ranging from1.5 to 8 M⊕ and 0.15 to 0.9 ρ⊕. Note that, even thoughtwo models are used to estimate stellar activities and giveconsistent estimates, the mass intervals might still evolvewith as more data comes along and the results become lessmodel-dependent. In particular, it seems like the mass ofplanet f, at 15.2 days, can be constrained despite activitysignatures at 16 days. A longer baseline would nonethe-less be suitable, so that the difference between 15.2 and 16would be greater than the frequency resolution.

6. Dynamics

6.1. A Laplace resonant chain

MMRs are orbital configurations where the period ratio ofa pair of planet is equal to, or oscillate around, a rationalnumber of the form (k+q)/k, where k and q are integers. InTOI-178, the candidates b and c are close to a second orderMMR (q = 2): Pc/Pb = 1.6915 ≈ 5/3, while c, d, e, f andg are pairwise close to first-order mean motion resonances(q = 1): Pd/Pc = 2.0249 ≈ 2/1, Pe/Pd = 1.5191 ≈ 3/2,Pf/Pe = 1.5290 ≈ 3/2 and Pg/Pf = 1.3595 ≈ 4/3. Pairs ofplanets lying just outside MMRs are common occurrencein systems observed by transit (Fabrycky et al. 2014). Tostudy such pair of planets, a relevant quantity is the dis-tance to the exact resonance. Taking the pair c and d asexample, the distance to the 2/1 MMR in the frequencyspace is given by 1nc− 2nd, where nc = 2π/Pc is the meanmotion of the planet c. Following Lithwick et al. (2012), wename ’super period’ the associated timescale:

Pc,d ≡1

|(k + q)/Pd − k/Pc|. (1)

The values of these quantities are given in Table 5 forthe planets b to g, along with the expression of the associ-



Table 6: Estimated values of the Laplace angles of planets c to g of TOI-178 toward the beginning of TESS Sector 2, at2458350.0BJD (August 2018). The derivative of the angles are averaged values between August 2018 and August 2020,based on the solution presented in tables 3 and 4. The equilibrium values of the Laplace angles are discussed in section6.2

ψj value [deg] dψj/dt [deg/year] equilibrium [deg] distance from eq. [deg]

ψ1 = φ1 − φ2 = 1λc − 4λd + 3λe 166.70+0.82−0.82 −4.74+0.47

−0.48 180 −13.30+0.82−0.82

ψ2 = φ2 − φ3 = 2λd − 5λe + 3λf 157.57+1.16−1.06 3.86+0.58

−0.60 168.94± 7.79 −11.37+7.88−7.86

ψ3 = 12 (φ3 − φ4) = 1λe − 3λf + 2λg 71.98+0.38

−0.48 0.15+0.23−0.22 78.90± 1.23 −6.92+1.29

−1.32

ated angles φi. TTVs are expected to happen over the superperiod, with amplitudes depending on the distance to theresonance, the mass of the perturbing planet and the ec-centricities of the pair (Lithwick et al. 2012). The fact thatthe super period of all three of these pairs are close to thesame value from planet c outward has additional implica-tions : The difference between the angles φi is evolving veryslowly. In other words, there is a Laplace relation betweenconsecutive triplets:

dψ1/dt = d(φ1 − φ2)/dt = 1nc − 4nd + 3ne ≈ 0 ,

dψ2/dt = d(φ2 − φ3)/dt = 2nd − 5ne + 3nf ≈ 0 ,

dψ3/dt =1

2d(φ3 − φ4)/dt = 1ne − 3nf + 2ng ≈ 0 ,

(2)

implying that the system is in a 2:4:6:9:12 Laplace resonantchain. The values of the Laplace angles ψj and derivativesare given in table 6. The values of the ψj are instantaneousand computed at the date 2458350.0 BJD which is towardthe beginning of the observation of TESS sector 2. As nosignificant TTVs were determined over the last two years,the derivatives of the ψj are average values over that period.The Laplace relations described in eq. (2) do not extendtoward the innermost triplet of the system: according toequations (C.3 and C.4), Pb,c should be equal to half Pc,dfor the b, c, d triplet to form a Laplace relation, which isnot the case, see table 5. To continue the chain, planet bwould have needed a period of ∼ 1.95 d. Its current periodof 1.91 d could indicate that it was previously in the chainbut pulled away, possibly by tidal forces.

Figure 9 shows the evolution of the Laplace angles whenintegrating the nominal solution given in Tables 3 and 4,starting at the beginning of the observation of TESS sec-tor 2. The three angles librate over the integrated time forthe selected initial conditions, with combination of peri-ods ranging from a few years to several decades. The ex-act periods and amplitudes of these variations depend onthe masses and eccentricities of the involved planets. Thelong-term stability of this system is discussed in section6.3, while the theoretical equilibria of the resonant anglesare discussed in section 6.2.

6.2. Equilibria of the resonant chain

For a given resonant chain there might exist several equilib-rium values around which the Laplace angles could librate(Delisle 2017). For instance, the four planets known to or-bit Kepler-223 are observed to librate around one of the sixpossible equilibria predicted by theory (Mills et al. 2016;Delisle 2017).

We use the method described in Delisle (2017) to deter-mine the position of the possible equilibria for the Laplace

angles of TOI-178. The five external planets (c to g) or-biting TOI-178 are involved in a 2:4:6:9:12 resonant chain.All consecutive pair of planets in the chain are close tofirst order MMR (1:2, 2:3, 2:3, 3:4). Moreover, as in theKepler-223 system, there are also strong interactions be-tween non-consecutive planets. Indeed, planet e and planetg (which are non-consecutive) are also close to a 1:2 MMR.As explained in Delisle (2017), this breaks the symmetryof the equilibria (i.e. the equilibrium is not necessarily at180 deg), and the position of the equilibria for the Laplaceangles depend on the planets’ masses.

We solved for the position of these equilibria using themasses given in Tables 3 and 4. We also propagated theerrors to estimate the uncertainty on the Laplace anglesequilibria. We find two possible equilibria for the system,which are symmetric from each other with respect to 0 deg.We provide the values of the Laplace angles correspondingto the first equilibrium in Table 6 (the second one is simplyobtained by taking ψj → −ψj for each angle).

It should be noted that these values correspond to theposition of the fixed point around which the system is ex-pected to librate. Depending on the libration amplitude,the instantaneous values of the Laplace angles can signifi-cantly differ from the equilibrium. For instance, in the caseof Kepler-223, the amplitude of libration could be deter-mined and is about 15 deg for all Laplace angles (Millset al. 2016). In Table 6, we observe that all instantaneousvalues of Laplace angles (as of August 2018) are also foundwithin 15 deg around the expected equilibrium. We notethat, during the two years of observations, ψ1 was movingaway from the equilibrium (see the averaged derivatives inTable 6). This would imply a final (as of September 2020)distance from equilibrium of about 23 deg. On the otherhand, ψ2 was getting closer to the equilibrium and ψ3 wasonly slowly evolving during this two years span. A more pre-cise determination of the evolution of Laplace angles (withmore data, and the detection of TTVs) would be needed toestimate the amplitude of libration of each angle. However,it is unlikely to find values so close to the equilibrium foreach of the three angles just by chance. Therefore, theseresults provide a strong evidence that the system is indeedlibrating around the Laplace equilibrium.

6.3. Stability

The planets c to g are embedded in a resonant chain, whichseems to greatly stabilize this planetary system.

The distance between the planets being quite small,their eccentricities may become a major source of insta-bility. This point is illustrated by figure 10, which shows asection of the system’s phase space. The initial conditionsand masses of the planets are those displayed in Tables 3



-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

ΔRV

[m/s

]

Mean Longitude [fraction of period]

RV data phase-folded at 1.91 days

-8

-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

ΔRV

[m/s

]

Mean Longitude [period fraction]


-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

ΔRV

[m/s

]



-6

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

ΔRV

[m/s

]



-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

ΔRV

[m/s

]



-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

ΔRV

[m/s

]



Fig. 8: Phase-folded RV. Error bars correspond to nominalerrors.

and 4 except that of the planet f , for which the initial pe-riod and eccentricity vary, while all other planets start oncircular orbits. The color code corresponds to a stabilityindex based on the diffusion of the main frequencies of the

0 20 40 60 80 100t [year]

50

75

100

125

150

175

200

225

250

reso

nant

ang

le [d

eg]

1

2

3

Fig. 9: Example of the evolution of the Laplace angles over100 years, starting from TESS observation of sector 2, usingthe masses and orbital parameters from Tables 3 and 4

system defined as (Laskar 1990, 1993):

log 10∣∣∣n(1)f − n

(2)f

∣∣∣ , (3)

where n(1)f and n(2)

f are the proper mean motion of planet fcomputed over the first half and the second half of the inte-gration, respectively, implying that the stability increasesfrom red to black (for more details see section 4.1 of Pe-tit et al. 2018). This map shows that the eccentricity ofplanet f must not exceed a few hundredths for the systemto remain stable. The same stability maps (not reproducedhere), made for each of the planets of the system, lead to thesame result: the planetary eccentricities have to be small inorder to guaranty the system’s stability. This constrain isverified if the system start with sufficiently small eccentric-ities. In particular, we verified that starting a set of numer-ical integration on circular orbit with masses and initialconditions close to the nominal one, and integrating thesystem over 100 000 years (i.e. about 19 million orbits ofthe planet b or 1.3 million orbits of planet g), do not excitethe eccentricities above one hundredth in most cases.

All the stability maps Pi vs ei, like the top panel ofFig. 10, show long quasi-vertical structures containing verystable regions in their central part and less stable regionsat their edges. These are mainly mean-motion resonancesbetween two or three planets. In particular, the blue areacrossed by the white dashed line in the top panel of Fig.10 corresponds to the resonant chain where the nominalsystem is located.

The bottom panel of Fig. 10 shows a different section ofthe phase space where Pf and Pg vary while all eccentric-ities are initialised at 0. This reveals part of the resonantstructure of the system. The blue regions are stable whilethe green to red areas mark the instability caused by theresonance web. This figure still highlights the stability is-land in which the planetary system is located. This narrowregion is surrounded by resonances: the ne− 3nf + 2ng = 0Laplace resonance (central diagonal strip), high order 2-body MMRs between planet f and g (horizontal strips);high order 2-body MMRs between planet e and f (verticallines).



The extent of this resonant chain versus planet f initialperiod and mass is shown in Fig. 11. On both panels, theX-axis corresponds to the orbital period of planet f , whilethe Y-axis corresponds to the planetary mass. The figurepresents two different, but complementary, aspects of thedynamics of TOI-178. The bottom panel indicates whetherthe Laplace angles ψ1, ψ2 and ψ3 librate or not. More pre-cisely, the color code corresponds to the number of anglesthat librate during the first 200 years of integration. Thetop panel present a stability index based on diffusion inmain frequencies. Three regions stand out clearly from thisfigure, each with a different dynamical regime.

The central region (yellow on the bottom panel) wherethe three Laplace angles librate simultaneously (see alsoFig.9) shows the heart of the 2:4:6:9:12 resonant chain,where the nominal system is located. On the stability map(top), its dark blue color reveals a very low diffusion rateand therefore a very long-term stability. This central regionseems not to dependent strongly on the mass of planet f .

On the other hand, the modification of the orbitalperiod has many more consequences. Indeed, outside ofthe central region where the three Laplace angles librates,which is about 0.015 days wide, chaotic layers are present(red on top panel panel, dark-blue on bottom) where noneof three Laplace angles librate. Here, the red color of thestability index corresponds to a significant, but moderatediffusion rate. Thus, although in this region the trajectoriesare not quasiperiodic, the chaos has limited consequences(it is bounded) and does probably not lead to the destruc-tion of the system.

Outside of these layers lies a very stable (quasiperiodic)region. The blue on the Laplace angle map shows that onlyone Laplace angle librates while the others circulate. Al-though the 2:4:6:9:12 chain is broken, the planets c, d, e, gremain inside the 2:4:6:12 resonant chain, independently ofplanet f orbital period. This demonstrates the robustness ofthese resonances. The stability map indicates a very strongregularity of the whole region. Nevertheless, one can noticethe presence of some narrow zones where the diffusion ismore important induced by high order orbital resonances,but without any significant consequence on the stability ofthe system.

Although the study described in this section gives onlya very partial picture of the structure of the phase space(parameter space) of the problem, it can be seen that aslong as the system is in the complete resonance chain, whichis the case for the nominal derived parameters and for thebulk of the posterior given in Tables 3 and 4, it remainsstable.

6.4. Expected TTVs

Since planets c, d, e, f and g are pairwise close to firstorder MMRs, we expect TTVs to occur over the super-period (Eq.1, see also Lithwick et al. 2012), which is roughly260 days for all these pairs (Table 5). The amplitude ofthese TTVs depends on the masses and eccentricities ofeach planet involved in the pair. Since the stability analysisconcludes that the system is more stable with eccentricitiesclose to 0, we integrate the initial conditions described inTables 3 and 4 over 6 years, starting during the observationof TESS sector 2 using the rebound package (Rein & Liu2012). The resulting TTVs are shown in Fig.12.

0 -1 -2 -3 -4 -5 -6

Stability indicator

e f

Pf [day]

0.00

0.02

0.04

0.06

0.08

0.10

15.10 15.15 15.20 15.25 15.30 15.35 15.40

P g [

day]

Pf [day]

20.55

20.60

20.65

20.70

20.75

20.80

20.85

20.90

15.10 15.15 15.20 15.25 15.30 15.35 15.40

Fig. 10: Stability indicator (defined in eq. 3) for TOI-178as a function of the periods and eccentricity of planet f(top), red shows the initial conditions of unstable trajecto-ries, while blue shows stable (quasi-periodic) trajectories.The bottom panel shows the same stability criterion as withrespect to the initial periods of planets f and g . The whitedashed lines show the observed periods reported in Table4.

In addition to the terms coming from the super-periods,the 6 years evolution of the TTVs shows the hints of thelong term evolution of the Laplace angles. As the exactshapes of the TTVs depends mainly on the masses of theinvolved planets, a yearly monitoring of the outer planets


A&A proofs: manuscript no. outputm

f [M

E]

Pf [day]

1

2

3

4

5

6

7

8

9

10

15.15 15.20 15.25 15.30

Fig. 11: Stability indicator as in Fig. 10 (top), and numberof librating Laplace angles (bottom) for TOI-178 as a func-tion of the period and mass of planet f . The white dashedline shows the observed period of planet f reported in Table4.

can bring precise constrains on the orbital configurationand masses of this system. The detailed study of the TTVsof this system will be the object of a forthcoming paper.

7. Internal structure

7.1. Minimum mass of the protoplanetary disk

The total mass of the planets detected in TOI-178 rangesfrom 17.37M⊕ to 29.76M⊕. Assuming a mass fraction of Hand He of maximum ∼ 20% (similar to the ones of Uranusand Neptune), the amount of heavy elements in the plan-ets is at least 13.8M⊕. This number can be compared withthe mass of heavy elements one would expect in a proto-planetary disk that would be similar to the Minimum MassSolar Nebula, but around TOI-178. Assuming a disk massequal to 1% of the stellar mass, and using the metallicity

of TOI-178 ([Fe/H] = 0.0061), the mass of heavy elementsin such a disk would be equal to ∼ 13M⊕, which is remark-ably similar to the minimum mass of heavy elements inplanets, as mentioned above. The concept of MMSN, scaledappropriately to reflect the reduced mass and metallicity ofTOI-178, seems therefore to be as applicable for this sys-tem as for the Solar System. Another implication of thiscomparison is that some problem in term of available masswould appear, should the planetary masses be revised tohigher values or should another massive planet be detectedin the same system. In such a case, the TOI-178 systemwould point towards a formation channel similar to the oneenvisioned for the Trappist-1 system (Schoonenberg et al.2019).

7.2. Mass-radius relation

The six planets we detected in the TOI-178 system are inthe super-Earth to mini-Neptune range, with radii rangingfrom 1.177±−0.074 to 2.91± 0.11 R⊕. Although the massdetermination is limited by the extent of the available spec-troscopic dataset, planet b and c appear to have roughly ter-restrial density 0.93+0.29

−0.33ρ⊕ and 0.95+0.18−0.21ρ⊕, respectively,

ρ⊕ being the density of the Earth. The outer planets seemto have a density significantly lower, in particular, we esti-mate the density of planet d to be 0.16+0.04

−0.05ρ⊕.Fig. 13 shows the position of the six planets in a mass-

radius diagram, in comparison with planets with mass andradius uncertainties less than 40% (light gray). Planets be-longing to four systems in Laplace resonance are indicatedon the same diagram: Trappist-1, Kepler-60, Kepler-80 andKepler 223. The diversity of planetary composition in TOI-178 is clearly visible on the diagram, with the two innerplanets having a radius compatible with a gas-free struc-ture, whereas the others contain water and/or gas. Thisis similar to the Kepler-80 system, where the two inner-most planets are compatible with a gas-free structure, andthe two outermost ones likely contain gas. Planets in theKepler-223, Kepler-60 and Trappist-1 seem to have a morehomogeneous structure, with all planets having a gas enve-lope in Kepler-223, and planets having a small gas envelopein Kepler-60 and Trappist-1.

Considering in more details the TOI-178 system, plan-ets d,e and g are located above the pure water line, they forsure contain a non-zero gas mass fraction. Planets d and,depending on its mass, planet g, are located in a part ofthe diagram where no other planets exist (at least no plan-ets with mass uncertainties smaller than 40 %) and mustcontain a large gas fraction.

Fig. 14 shows the location of the TOI-178 planets in astellar isolation versus planetary radius diagram. The colorcode illustrates the density of exoplanets. This diagramsclearly shows the so-called ’evaporation valley’4. Planets band c are located below the valley, and their high densitycould result from the evaporation of a primordial envelope.The outer planets are located above the valley, and haveprobably preserved (part of) their primordial gas envelope.The present day architecture of the TOI-178 system cantherefore be emphasized considering the densities of planets

4 Note that if the presence of a valley seems robust, it couldbe due to effect that are not related to evaporation, e.g. corecooling (Gupta & Schlichting 2019) or from combined formationand evolution effects (Venturini et al. 2020).



Fig. 12: Example of TTVs of the 6 planets of TOI-178, starting from TESS observation of Sector 2 and spanning 6 years,using the masses and orbital parameters from Tables 3 and 4.

Fig. 13: The TOI-178 planets compared to other known transiting exoplanets with radius and mass uncertainties lessthan 40% (grey), and other system known to harbour a Laplace resonance. Data on known exoplanets were taken fromNASA Exoplanet Archive on 18 September 2020. The dashed lines show theoretical mass-radius curves for some idealizedcompositionsZeng et al. (2019). The six planets orbiting around TOI-178 are indicated, the color of the points and errorbars giving the equilibrium temperature. The seven planets orbiting Trappist-1 are shown with diamonds, the parametersbeing taken from Agol et al. (2020). The three planets orbiting Kepler-60 are shown with X, the parameters are takenfrom Jontof-Hutter et al. (2016). The four planets orbiting Kepler-80 are shown with bottom pointing triangles, theparameters being taken from MacDonald et al. (2016). The four planets orbiting Kepler-223 are shown with up pointingtriangles, the parameters being taken from Mills et al. (2016).



Fig. 14: Position of the TOI-178 planets in a stellar light in-tensity (relative to Earth) versus planetary radius diagram.The marker size is inversely proportional to the density andthe color-code gives the density of exoplanets, from yellowfor empty regions of the diagram to violet for high-density/ highly populated regions.

which is known to results from combined formation andevaporation processes.

7.3. Comparison with other systems in Laplace resonance

The difference between the TOI-178 system and other sys-tems in Laplace resonance can clearly be seen on Fig. 15where we show the density of planets as a function of theirequilibrium temperature for the same systems as in Fig.13. In Kepler-60, Kepler-80 and Kepler-223, the density ofplanets is decreasing when the equilibrium temperature de-creases. This can be understood as the effect of evapora-tion that removes part of the primordial gaseous envelope,this effect being stronger for planets closer to their star. InTrappist-1, the density of planets is always higher than 4g/cm3, and increasing (with the exception of Trappist-1 f)when decreasing the equilibrium temperature. This is likelyto result from the presence of more ices in planets far fromthe star (Agol et al. 2020). In the TOI-178 system, the den-sity of planets is not a growing function of the equilibriumtemperature, as for the three Kepler systems. Indeed, TOI-178f has a density higher than planets e , and TOI-178dhas a density smaller than planet e.

Planet f is substantially more massive than all the otherplanets in the system. From a formation perspective, onewould expect that this planet should have a density smallerthan the other planets, in particular planet e, at the endof the formation phase, as we can observe for example forJupiter, Saturn, Uranus and Neptune. Since this planet isfurther away from the star compared to planet e, evapora-tion should have been less effective for planet f comparedto planet e. The combined effect of formation and evolutionshould therefore lead to a smaller density for planet f com-pared to planet e. Similarly, planet d is smaller than planete and located closer to the star. Using the same arguments,the combined effect of formation and evolution should haveled to planet d having a density larger than planet e. TheTOI-178 system seems therefore at odd with the general un-derstanding of planetary formation and evaporation whereone would expect the density to decrease when the distanceto the star increase, or when the mass of the planet increase.

Fig. 15: Densities of planets in the TO-178, Trappist-1,Kepler-60, Kepler-80 and Kepler-223 systems, as a func-tion of their equilibrium temperature. The error bars givethe 16% and 84% quantile and the marker is located at themedian of the computed distribution. The color code givesthe planetary mass in Earth masses. The parameters of theplanets are taken from the references mentioned in Fig. 13.

7.4. Internal structure modeling

We have used a Bayesian analysis in order to compute theposterior distribution of the planetary internal structureparameters. The method we use follows closely the one ofDorn et al. (2015) and Dorn et al. (2017), and has alreadybeen used in Mortier et al. (2020) and Delrez (2020 - sub-mitted). We here recall the main physical assumptions ofthe model.

The model is split in two parts, the first is the for-ward model, which provide the planetary radius as a func-tion of the internal structure parameters, the second is theBayesian analysis which provide the posterior distributionof the internal structure parameters, given the observedradii, masses, and stellar parameters (in particular its com-position).

For the forward model, we assume that each planet ismade of four layers: an iron/sulfur inner core, a mantle, awater layer and a gas layer. We use for the core the Equa-tion of State (EOS) of Hakim et al. (2018), for the silicatemantle, the EOS of Sotin et al. (2007), and the water EOS istaken from Haldemann et al. (2020). These three layers con-stitute the ’solid’ part of the planets. The thickness of thegas layer (assumed to be made of pure H/He) is computedas a function of the stellar age, mass and radius of the solidpart, and irradiation from the star, using the formulas ofLopez & Fortney (2014). The internal structure parametersof each planet are therefore the iron molar fraction in thecore, the Si and Mg molar fraction in the mantle, the massfraction of all layers (inner core, mantle, water), the age ofthe planet (equal to the age of the star) and the irradiationfrom the star. More technical details on the calculation ofthe forward model are given in Appendix D.



In the Bayesian analysis part of model, we proceed intwo steps. We first generate 150000 synthetic stars, theirmass, radius, effective temperature, and age being taken atrandom following the stellar parameters computed in Sect.3. The Fe/Si/Mg bulk molar ratio in the star are assumedto be solar, with an uncertainty of 0.05 (uncertainty on[Fe/H], see 3). For each of these stars, we generate 1000planetary systems, varying the internal structure parame-ters of all planets, and assuming that the bulk Fe/Si/Mgmolar ratios are equal to the stellar ones. We then com-pute the transit depth and RV semi-amplitude for each ofthe planets, and retain models that fit the observed datawithin the error bars. By doing so, we include the fact thatall synthetic planets orbit a star with exactly the same pa-rameters. Indeed, planetary masses and radii are correlatedby the fact that the fitted quantities are the transit depthand RV semi-amplitude, which depend on the stellar radiusand mass. In order to take into account this correlation, itis therefore important to fit the planetary system at once,and not each planet independently.

For the Bayesian analysis, we assume the following pri-ors: the mass fraction of the gas envelop is assumed to beuniform in log. For the solid part, the mass fraction of theinner core, mantle and water layer are uniform on the sim-plex (the surface on which they add up to one). We assumein addition that the mas fraction of water is smaller than 50% (Thiabaud et al. 2014; Marboeuf et al. 2014). The molarfraction of iron in the inner core is uniform between 0.5 and1, and the molar fraction of Si, Mg and Fe in the mantleis uniform on the simplex (they also add up to one). In or-der to compare TOI-178 with other systems with a Laplaceresonance, we have also done a Bayesian analysis for theKepler-60, Kepler-80 and Kepler-223 systems in order tocompute the probability distribution of the gas mass in thedifferent planets. The parameters for all systems are takenfrom the references mentioned above, and for all systemswe have assumed that [Si/H]=[Mg/H]=[Fe/H]. We have notconsidered the Trappist-1 system in this comparison, as it islikely that the variations in density in these planets resultsfrom variation in their ice content (Agol et al. 2020).

The posteriors distributions of the most important pa-rameters (mass fractions, composition of the mantle) ofeach planets in TOI-178 are shown in Appendix D, Fig.D.2 to D.7. We focus here on the mass of gas in each of theplanet, and plot in Fig. 16 the mass of the gaseous envelopfor each planet as a function of their equilibrium tempera-ture.

As can be seen on Fig. 16, the gas mass in planets gener-ally decreases when the equilibrium temperature decreasesin all three Kepler systems. One exception to this tendencyis mass of Kepler-223d’s envelope which is larger than theone of planet e in the same system. Kepler-223d is howevermore massive than planets c and e in the same system, andone expect from formation models that the mass of the pri-mordial gas envelope is a growing function of the total plan-etary mass. In the case of TOI-178, the mass of gas globallyalso increases when the equilibrium temperature decrease,with the notable exception of planet d. Indeed, a linearinterpolation would provide for planet d a gas mass of theorder of 10−6−10−5M⊕ whereas the interior structure mod-elling gives a value of 9.66 × 10−3M⊕ and 2.56 × 10−2M⊕for the 16% and 84% quantiles. Even more intriguing, ourresults show that the amount of gas in planet d is largerthan in planet e, this latter being both more massive and

Fig. 16: Gas mass in the planets of the TOI-178, Kepler-60,Kepler-80 and Kepler-223 systems, as a function of theirequilibrium temperature. The error bars give the 16% and84% quantile and the marker is located at the median of thecomputed distribution. The color code gives the planetarymass in Earth masses.The parameters of the planets aretaken from the references mentioned in Fig. 13.

at larger distance from the star. Indeed, from the joint prob-ability distribution of all planetary parameters as providedby the Bayesian model, the probability that planet d hasmore gas than planet e is equal to 92%.

From a formation point of view, one generally expectthat the mass of gas is a growing function of the core mass.From an evolution perspective, one also expect that evap-oration should be more effective for planets closer to thestar. Both would point towards a gas mass in planet dthan should be smaller than the one in planet e. The largeamount of gas in planet d, and more generally the apparentirregularity in the planetary envelope masses, is surprisingin view of the apparent regularity of the orbital configura-tion of the system that was presented in Sect. 6.

8. Discussion and conclusions

In this study we presented new observations of TOI-178 byCHEOPS, ESPRESSO, NGTS and SPECULOOS. Thanksto this follow-up effort we were able to solve the architectureof the system: out of the three previously announced can-didates at 6.56 d, 9.96 d and 10.3 d, we confirm the first two(6.56 d and 9.96 d) and re-attribute the transits of the thirdto 15.23 d and 20.7 d planets, in addition to the detectionof two new inner planets at 1.91 d and 3.24 d, all of theseplanets were confirmed by follow-up observations. In total,we hence announce six planets in the super-Earth to mini-Neptune range, with orbital periods from 1.9 d to 20.7 d,all of them but the innermost in a 2:4:6:9:12 Laplace res-onances chain. Current ephemerides and mass estimationsindicate a very stable system, with Laplace angles libratingover decades.

As there is no theoretical reason for the resonantchain to stop at 20.7 days, and the current limit probablycomes from the duration of the available photometric and



Table 7: Potential periods that would continue the resonantchain in a near (k + q) : k MMR with planet g, takingthe super-period at 260 days, see equations in Appendix C.Changing the super-period by a few days typically changethe resulting period by less than 0.1 day.

k, q Period [day]

1, 1 45.00282, 1 32.35223, 1 28.36534, 1 26.41245, 1 25.25333, 2 36.45085, 2 29.94707, 2 27.2461

radial velocity datasets, we give in table 7 the periodsthat continue the resonant chain for first (q = 1) andsecond order (q = 2) MMRs. These periods result from theequations detailed in Appendix C. In the TOI-178 system,as well as in similar systems in Laplace resonance, planetpairs are nearly all near first order MMRs. The most likelyof the periods shown in table 7 are therefore the first ordersolutions with low k hence 45.00, 32.35, or 28.36 days. Wenote that, for a star like TOI-178, the inner boundary ofthe habitable zone lies around 0.2 AU, or at a period ofthe order of 40 days. Additional planets in the Laplaceresonance could therefore orbit inside, or very close to, thehabitable zone.

The brightness of TOI-178 allows for further character-isation of the system both by photometric measurements,radial velocities and transit spectroscopy. These measure-ments will be essential to further constrain the system, notonly on its orbital architecture, but also for the physicalcharacterisation of the different planets.

As discussed above, the current mass and radii deter-minations show significant differences between the compo-sitions of the different planets. It appears that the two in-nermost planets are likely to be rocky, which may be dueto the fact that they have lost their primary, hydrogen-dominated atmospheres through escape (Kubyshkina et al.2018, 2019), whereas all the other planets may have re-tained part of their primordial gas envelope. In this respect,the different planets of the TOI-178 system lie on bothsides of the radius valley (Fulton et al. 2017). Therefore,reconstructing the past orbital and atmospheric history ofthis planet may provide clues regarding the origin of thevalley. The system is located at a declination in the skythat makes it observable by most ground-based observato-ries around the globe. Furthermore, the host star is brightenough and the radii of the outer planets large enough tomake them possibly amenable to optical and infrared trans-mission spectroscopy observations from both ground andspace, particularly by employing the upcoming E-ELT andJWST facilities. Indeed, it has been shown that the JamesWebb Space Telescope has the capacity to perform trans-mission spectroscopy of planets with radii down to 1.5 R⊕(Samuel et al. 2014).

The two planets d and f are particularly interesting astheir density if very different from the ones of their neigh-

bors, and as they depart from the general tendency of plan-etary density decreasing for decreasing equilibrium temper-atures. The densities of planest d and f , in the context ofthe general trend seen in TOI-178, is difficult to understandin term of formation and evaporation process, and could bedifficult to reproduce by planetary system formation mod-els (Mordasini et al. 2012; Alibert et al. 2013; Emsenhuberet al. 2020). We stress that, even though two different anal-yses yielded similar estimates of mass and densities, theyare made with only forty-six data points. As such, these es-timates need to be confirmed by further RV measurements,which would provide in particular a better frequency reso-lution and confirm the mass estimate of planet f.

We note finally that the orbital configuration of TOI-178is too fragile to survive giant impacts, or even significantclose encounters: Fig. 10 shows that sudden change of pe-riod of one of the planets of less than a few ∼ .01 d canrender the system chaotic, while Fig. 11 shows that modi-fying a single period axis can break the resonant structureof the entire chain. Understanding in a single frameworkthe apparent dis-order in term of planetary density on oneside, and the very high level of order seen in the orbitalarchitecture on the other side will be represent a challengefor planetary system formation models. Additional obser-vations with CHEOPS and RV facilities will allow furtherconstraining the internal structure of all planets in the sys-tem, and in particular the (lack of) similarity between thewater and gas mass fraction between planets.

In addition, TTVs are expected to be large in the TOI-178 system (see Fig. 12), and to occur on periods of years.Future observations of this system can be used to measurethe planetary masses directly from TTVs, and compare theresult with masses derived from RV measurements. Thiscould provide a benchmark to assess the capability of massmeasurement through TTVs.

Furthermore, the innermost planet, b, lies just outsidethe 3:5 MMR with planet c, albeit a bit too far to bepart of the Laplace chain, which would require a periodof ∼ 1.95d. Since the formation of the Laplace resonantchain probably result from a slow drift from a chain oftwo-planet resonances due to tidal effects (Papaloizou &Terquem 2010; Delisle et al. 2012; Papaloizou 2015; Mac-Donald et al. 2016), the current state of the system mightconstrain the dissipative processes that tore apart the in-nermost link of the chain, while the rest of the configurationsurvived.

The TOI-178 system as revealed by the recent observa-tions described in this paper cumulates a number of veryimportant features: Laplace resonances, variation in densi-ties from planet to planet, stellar brightness that allowsa number of follow-up observations (photometric, atmo-spheric, spectroscopic). It is therefore likely to become oneof the Rosetta Stones to understand planet formation andevolution, even more so if additional planets continuing thechain of Laplace resonances would be discovered orbitinginside the habitable zone.

Software list:

– allesfitter (Günther & Daylan 2020).



– astropy.– batman/– celerite (Foreman-Mackey et al. 2017a).– CHEOPS DRP (Hoyer et al. 2020).– CHEOPSim (Futyan et al. 2020).– dynesty (Speagle 2020; Skilling 2004, 2006a; Higson

et al. 2019; Buchner 2014, 2017; Skilling 2006b).– ellc (Maxted 2016).– emcee (Foreman-Mackey et al. 2013).– ipython (Pérez & Granger 2007).– `1 periodogram (Hara et al. 2017)5– lmfit (Newville et al. 2014).– matplotlib (Hunter 2007).– numpy (Harris et al. 2020).– pycheops6 Maxted et al., in prep..– rebound– seaborn7

– scipy (Virtanen et al. 2020).– spleaf (Delisle et al. 2020)8– tqdm (da Costa-Luis et al. 2018).– Tensorflow– Keras

Acknowledgements. The authors acknowledge support from the SwissNCCR PlanetS and the Swiss National Science Foundation. YA andMJH acknowledge the support of the Swiss National Fund under grant200020_172746. ACC and TW acknowledge support from STFCconsolidated grant number ST/M001296/1. This work was grantedaccess to the HPC resources of MesoPSL financed by the Region Ile deFrance and the project Equip@Meso (reference ANR-10-EQPX-29-01)of the programme Investissements d’Avenir supervised by the AgenceNationale pour la Recherche. Based on data collected under theNGTS project at the ESO La Silla Paranal Observatory. The NGTSfacility is operated by the consortium institutes with support fromthe UK Science and Technology Facilities Council (STFC) projectST/M001962/1. V.A. acknowledges the support from FCT throughInvestigador FCT contract nr. IF/00650/2015/CP1273/CT0001.We acknowledge support from the Spanish Ministry of Scienceand Innovation and the European Regional Development Fundthrough grants ESP2016-80435-C2-1-R, ESP2016-80435-C2-2-R,PGC2018-098153-B-C33, PGC2018-098153-B-C31, ESP2017-87676-C5-1-R, MDM-2017-0737 Unidad de Excelencia “María de Maeztu”-Centro de Astrobiología (INTA-CSIC), as well as the support ofthe Generalitat de Catalunya/CERCA programme. The MOCactivities have been supported by the ESA contract No. 4000124370.S.C.C.B. acknowledges support from FCT through FCT contractsnr. IF/01312/2014/CP1215/CT0004. XB, SC, DG, MF and JLacknowledge their role as ESA-appointed CHEOPS science teammembers. ABr was supported by SNSA. A.C. acknowledges supportby CFisUC projects (UIDB/04564/2020 and UIDP/04564/2020),ENGAGE SKA (POCI-01-0145-FEDER-022217), and PHOBOS(POCI-01-0145-FEDER-029932), funded by COMPETE 2020 andFCT, Portugal. The Belgian participation to CHEOPS has been sup-ported by the Belgian Federal Science Policy Office (BELSPO) in theframework of the PRODEX Program, and by the University of Liègethrough an ARC grant for Concerted Research Actions financed bythe Wallonia-Brussels Federation. This work was supported by FCT -Fundação para a Ciência e a Tecnologia through national funds and byFEDER through COMPETE2020 - Programa Operacional Competi-tividade e Internacionalização by these grants: UID/FIS/04434/2019;UIDB/04434/2020; UIDP/04434/2020; PTDC/FIS-AST/32113/2017POCI-01-0145-FEDER- 032113; PTDC/FIS-AST/28953/2017POCI-01-0145-FEDER-028953; PTDC/FIS-AST/28987/2017POCI-01-0145-FEDER-028987. O.D.S.D. is supported in the formof work contract (DL 57/2016/CP1364/CT0004) funded by nationalfunds through FCT. B.-O.D. acknowledges support from the SwissNational Science Foundation (PP00P2-163967). MF and CMP grate-fully acknowledge the support of the Swedish National Space Agency(DNR 65/19, 174/18). DG gratefully acknowledges financial support

5 https://github.com/nathanchara/l1periodogram6 https://github.com/pmaxted/pycheops7 https://seaborn.pydata.org/index.html8 https://gitlab.unige.ch/Jean-Baptiste.Delisle/spleaf

from the CRT foundation under Grant No. 2018.2323 “Gaseousorrocky? Unveiling the nature of small worlds”. EG gratefully acknowl-edges support from the David and Claudia Harding Foundation inthe form of a Winton Exoplanet Fellowship. M.G. is an F.R.S.-FNRSSenior Research Associate. MNG ackowledges support from the MITKavli Institute as a Juan Carlos Torres Fellow. JH acknowledges thesupport of the Swiss National Fund under grant 200020_172746.JSJ acknowledges support by FONDECYT grant 1201371, andpartial support from CONICYT project Basal AFB-170002. AJacknowledges support from ANID – Millennium Science Initiative –ICN12_009 and from FONDECYT project 1171208. This work wasalso partially supported by a grant from the Simons Foundation (PIQueloz, grant number 327127). Acknowledges support from the Span-ish Ministry of Science and Innovation and the European RegionalDevelopment Fund through grant PGC2018-098153-B- C33, as wellas the support of the Generalitat de Catalunya/CERCA programme.S.G.S. acknowledge support from FCT through FCT contractnr. CEECIND/00826/2018 and POPH/FSE (EC). This researchreceived funding from the MERAC foundation, from the EuropeanResearch Council under the European Union’s Horizon 2020 researchand innovation programme (grant agreement nº 803193/ BEBOP,and from the Science and Technology Facilites Council (STFC,grant nº ST/S00193X/1). V.V.G. is an F.R.S-FNRS Research Asso-ciate. JIV acknowledges support of CONICYT-PFCHA/DoctoradoNacional-21191829. M. R. Z. O. acknowledges financial supportfrom projects AYA2016-79425-C3-2-P and PID2019-109522GB-C51from the Spanish Ministry of Science, Innovation and Universities.This work has made use of data from the European Space Agency(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processedby the Gaia Data Processing and Analysis Consortium (DPAC,https://www.cosmos.esa.int/web/gaia/dpac/consortium). Fundingfor the DPAC has been provided by national institutions, in particularthe institutions participating in the Gaia Multilateral Agreement.

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1 Observatoire Astronomique de l’Université de Genève, Chemindes Maillettes 51, Sauverny, CH-1290, Switzerland2 Physikalisches Institut, University of Bern, Gesellsschaft-strasse 6, 3012 Bern, Switzerland3 Centre for Exoplanet Science, SUPA School of Physicsand Astronomy, University of St Andrews, North Haugh, StAndrews KY16 9SS, UK4 IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ.,



Sorbonne Univ., 77 av. Denfert-Rochereau, 75014 Paris, France5 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France6 Department of Physics, University of Warwick, Gibbet HillRoad, Coventry CV4 7AL, UK7 Centre for Exoplanets and Habitability, University of War-wick, Gibbet Hill Road, Coventry CV4 7AL, UK8 Astrobiology Research Unit, Université de Liège, Allée du 6Août 19C, B-4000 Lige, Belgium9 Institute of Planetary Research, German Aerospace Center(DLR), Rutherfordstrasse 2, 12489 Berlin, Germany10 School of Physics and Astronomy, University of Leicester,Leicester LE1 7RH, UK11 Instituto de Astrofísica e Ciências do Espaço, Universidadedo Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal12 Centro de Astrofísica da Universidade do Porto, Rua dasEstrelas, 4150-762 Porto, Portugal13 Departamento de Física e Astronomia, Faculdade de Ciên-cias, Universidade do Porto, Rua do Campo Alegre, 4169-007Porto, Portugal14 Instituto de Astrofísica de Canarias, 38200 La Laguna,Tenerife, Spain15 Departamento de Astrofísica, Universidad de La Laguna,38206 La Laguna, Tenerife, Spain16 Camino El Observatorio 1515, Las Condes, Santiago, Chile17 ETH Zürich, Institute for Particle Physics and Astrophysics18 Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB,Can Magrans s/n, 08193 Bellaterra, Spain19 Institut d’Estudis Espacials de Catalunya (IEEC), 08034Barcelona, Spain20 ESTEC, European Space Agency, 2201AZ, Noordwijk, NL21 Depto. de Astrofísica, Centro de Astrobiologia (CSIC-INTA),ESAC campus, 28692 Villanueva de la Cãda (Madrid), Spain22 Space Research Institute, Austrian Academy of Sciences,Schmiedlstrasse 6, A-8042 Graz, Austria23 Center for Space and Habitability, Gesellsschaftstrasse 6,3012 Bern, Switzerland24 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble,France25 Department of Astronomy, Stockholm University, AlbaNovaUniversity Center, 10691 Stockholm, Sweden26 Institute of Optical Sensor Systems, German AerospaceCenter (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany27 Department of Earth, Atmospheric and Planetary Science,Massachusetts Institute of Technology, 77 MassachusettsAvenue, Cambridge, MA 02139, USA28 Admatis, Miskok, Hungary29 Université de Paris, Institut de physique du globe de Paris,CNRS, F-75005 Paris, France30 CFisUC, Department of Physics, University of Coimbra,3004-516 Coimbra, Portugal31 INAF - Osservatorio Astronomico di Trieste, via G. B.Tiepolo 11, I-34143 Trieste, Italy32 INAF - Osservatorio Astrofisico di Torino, via Osservatorio20, 10025 Pino Torinese, Italy33 Lund Observatory, Dept. of Astronomy and TheoreticalPhysics, Lund University, Box 43, 22100 Lund, Sweden34 Astrobiology Research Unit, Université de Liège, Allée du 6Août 19C, 4000 Liège, Belgium35 Space sciences, Technologies and Astrophysics Research(STAR) Institute, Université de Liège, Allée du 6 Août 19C,4000 Liège, Belgium36 INAF, Istituto di Astrofisica e Planetologia Spaziali, via delFosso del Cavaliere 100, 00133 Roma, Italy37 European Southern Observatory, Alonso de Coórdova 3107,Vitacura, Regioón Metropolitana, Chile38 Leiden Observatory, University of Leiden, PO Box 9513,2300 RA Leiden, The Netherlands39 Department of Space, Earth and Environment, ChalmersUniversity of Technology, Onsala Space Observatory, 43992Onsala, Sweden

40 Dipartimento di Fisica, Università degli Studi di Torino, viaPietro Giuria 1, I-10125, Torino, Italy41 Center for Astronomy and Astrophysics, Technical UniversityBerlin, Hardenberstrasse 36, 10623 Berlin, Germany42 Astronomy Unit, Queen Mary University of London, MileEnd Road, London E1 4NS, UK43 Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB30HE, UK44 University of Vienna, Department of Astrophysics, Türken-schanzstrasse 17, 1180 Vienna, Austria45 Department of Physics and Kavli Institute for Astrophysicsand Space Research, Massachusetts Institute of Technology,Cambridge, MA 02139, USA46 Space Sciences, Technologies and Astrophysics Research(STAR) Institute, Université de Liège, Allée du 6 Août 19C,B-4000 Liège, Belgium47 Departamento de Astronomía, Universidad de Chile, CaminoEl Observatorio 1515, Las Condes, Santiago, Chile48 Centro de Astrofísica y Tecnologías Afines (CATA), Casilla36-D, Santiago, Chile49 Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez,Av. Diagonal las Torres 2640, Peñalolén, Santiago, Chile50 Millennium Institute for Astrophysics, Chile51 Konkoly Observatory, Research Centre for Astronomy andEarth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17,Hungary52 Brorfelde Observatory, Observator Gyldenkernes Vej 7,DK-4340 Tølløse, Denmark53 DTU Space, National Space Institute, Technical Universityof Denmark, Elektrovej 327, DK-2800 Lyngby, Denmark54 Institut d’astrophysique de Paris, UMR7095 CNRS, Uni-versité Pierre & Marie Curie, 98bis blvd. Arago, 75014 Paris,France55 INAF, Osservatorio Astronomico di Padova, Vicolodell’Osservatorio 5, 35122 Padova, Italy56 Astrophysics Group, Keele University, Staffordshire, ST55BG, United Kingdom57 Department of Physics, University of Warwick, Coventry,UK58 INAF - Osservatorio Astronomico di Palermo, Piazza delParlamento 1, 90134, Palermo, Italy59 IFPU, Via Beirut 2, 34151 Grignano Trieste60 Instituto de Astronomía, Universidad Católica del Norte,Angamos 0610, Antofagasta, Chile61 Instituto de Astrofísica e Ciências do Espaço, Faculdadede Ciências da Universidade de Lisboa, Campo Grande,PT1749-016 Lisboa, Portugal62 Department of Astrophysics, University of Vienna, Tuerken-schanzstrasse 17, 1180 Vienna, Austria63 INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78,95123 Catania, Italy64 Instituto de Astrofísica de Canarias, 38200 La Laguna,Tenerife, Spain65 Departamento de Astrofísica, Universidad de La Laguna,38206 La Laguna, Tenerife, Spain66 Dipartimento di Fisica e Astronomia "Galileo Galilei",Università degli Studi di Padova, Vicolo dell’Osservatorio 3,35122 Padova, Italy67 Space Science Data Center, ASI, via del Politecnico snc,00133 Roma, Italy68 Department of Physics, University of Warwick, Gibbet HillRoad, Coventry CV4 7AL, United Kingdom69 Fundación G. Galilei – INAF (Telescopio Nazionale Galileo),Rambla J. A. Fernández Pérez 7, E-38712 Breña Baja, LaPalma, Spain70 INAF - Osservatorio Astronomico di Brera, Via E. Bianchi46, I-23807 Merate, Italy71 Institut für Geologische Wissenschaften, Freie UniversitätBerlin, 12249 Berlin, Germany72 Paris Observatory, LUTh UMR 8102, 92190 Meudon, France



73 School of Physics Astronomy, University of Birmingham,Edgbaston, Birmingham B15 2TT, United Kingdom74 ELTE Eötvös Loránd University, Gothard AstrophysicalObservatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary75 MTA-ELTE Exoplanet Research Group, 9700 Szombathely,Szent Imre h. u. 112, Hungary76 INAF, Osservatorio Astronomico di Roma, via Frascati 33,00078 Monte Porzio Catone, Roma, Italy77 Institute of Astronomy, University of Cambridge, MadingleyRoad, Cambridge, CB3 0HA, United Kingdom78 Centro de Astrobiología (CSIC-INTA), Crta. Ajalvir km 4,E-28850 Torrejón de Ardoz, Madrid, Spain

Appendix A: Inspection of the CHEOPS data

The four CHEOPS visits were automatically processedthrough the DRP with individual frames undergoing vari-ous calibrations and corrections, with aperture photometrysubsequently conducted for four aperture radii as has beenhighlighted in Section 4.1.2 and covered in detailed in Hoyeret al. 2020. The produced light curves with, often referredto as “raw” in order to indicate no post-processing detrend-ing has taken place, obtained with the DEFAULT aperture,for all runs in this study are shown in Figs. A.1. For thefirst, third, and fourth visits the standard data processingwithin the DRP was performed, however for the second runcareful treatment of telegraphic pixels was needed.

In CHEOPS CCD there is a large number of hot pix-els (see for example Fig. 3). Moreover, some normal pixelscan change their behaviour to an abnormal state within theduration of a visit. For example, a pixel can become hot af-ter a SAA crossing of the satellite. These pixels are calledtelegraphic due to their unstable response during the obser-vations; and they can disturb the photometry if located in-side the photometric aperture. To discard that the detectedtransit events in the light curves correspond to the effect oftelegraphic pixels, the data frames were carefully inspectedand compared versus the detection map of hot pixels de-livered by the CHEOPS DRP (see details in Hoyer et al.2020). By doing this, one telegraphic pixel was detected in-side the DEFAULT aperture at the end of the second visitof TOI-178. The exact CCD location of this abnormal pixelis shown if Fig. 3. The effect of this pixel in the photometryis shown in the form of a jump in flux in the light curve ofthe visit (top panel, Fig. A.2) at BJD∼2 459 076.5. whichcorresponds to the flux increase of the pixel (middle panel,Fig. A.2). After correcting the data, by simply cancellingthe flux of this pixel through the full observation, and re-peating the photometry extraction with the same aperture(R=25′′) we removed the flux jump in the light curve (bot-tom panel, Fig. A.2). No telegraphic pixels were detectedon the rest of the visits of TOI-178.

Appendix B: Analysis of the radial velocity data

Appendix B.1: Method

In this appendix, we describe the analysis of the RV dataalone. To search for planet detections, we computed the `1periodogram of the RV, as defined in Hara et al. (2017).This tool is based on a sparse recovery technique called thebasis pursuit algorithm (Chen et al. 1998). It aims at findinga representation of the RV time series as a sum of a smallnumber of sinusoids whose frequencies are in the input grid.

Fig. A.1: The DRP produced light curves (DEFAULT aper-ture) of the 4 CHEOPS visits of TOI-178 presented in thiswork. 3-σ outliers have been removed for better visualiza-tion.

Fig. A.2: TOI-178 normalized light curve of the second visit(gray symbols) with its 10min smoothed version overplot-ted (blue symbols) is shown in the top panel. The flux jumpsproduced by the appearance of a new hot pixel are markedwith the dashed vertical lines. The light curve of the de-tected telegraphic pixel is presented in the middle panelshowing its anomalous behaviour at the end of the visit. Thelight curve extracted from the corrected data is shown inthe bottom panel. For better visualization, the light curvesare presented after a 3-σ clipping and corrected by a secondorder polynomial in time.



The `1 periodogram presents the advantage, over a regularperiodogram, to search for several periodic components atthe same time, and therefore drastically reduces the impactof aliasing (Hara et al. 2017).

The `1 periodogram has three inputs: a frequency grid,onto which the signal will be decomposed in the Fourierdomain, a noise model in the form of a covariance matrix,and a base model. The base model represents offsets, trends,or activity models, and can be understood as follows. Theprinciple behind the `1 periodogram is to consider the signalin the Fourier domain, to minimise the sum of the absolutevalue of the Fourier coefficients on a discretized frequencygrid (their `1 norm), while ensuring that the inverse Fouriertransform is close enough to the model in a certain, precisesense. Due to the `1 norm penalty, frequencies “compete”against each other to have non zero coefficients. However,one might assume that certain frequencies, and more largelycertain signals are by default in the data, and should notbe penalized by the `1 norm. We can define a linear modelwhose column vectors are not penalized, which we call thebase model.

The signals found to be statistically significant mightvary depending on the frequency grid, base and noise mod-els. To explore this aspect, as in Hara et al. (2020), wecompute the `1-periodogram of the data with different as-sumptions on the noise covariance. The covariance modelsare then ranked via cross-validation. That is, we fix a fre-quency grid. Secondly, for every choice of base and noisemodels, we record which detections are announced. We as-sess the score of the detections + noise models by crossvalidation. The data is separated in a training and test setcontaining respectively 70 and 30 % of the data, chosen atrandom. The model is fitted onto the training set and onecomputes the likelihood of the data on the test set. Thisoperation is repeated 250 times, and we attribute the me-dian of the 250 scores to the triplet base model, covariancemodel and signal detected. To determine if a signal at agiven period is significant, we study the distribution of itsFAP among the highest ranked models.

Appendix B.2: Definition of the alternative models

To define the alternative RV models, as a preliminary step,we analyse the Hα, FWHM, bisector span and logR′HKtime-series as provided by the ESPRESSO pipeline. Wecompute the residual periodograms, as described in (Baluev2008). These periodograms allow to take into account gen-eral linear base models that are fitted along candidate fre-quencies. We compute the periodgrams and iteratively adda sinusoidal function whose frequency is corresponding themaximum peak of the periodogram. Iterations 1 and 2 areshown in Fig. B.1, respectively top and bottom. We markwith the dotted black the position of 36 days and 16 days(top and bottom, respectively). Pursuing the iterations, wefind signals detection with FAP <10−3 of periods at 36,115 and 15.9 for Hα, 35.5, 20.8 and 145 for the FWHM,36 and 16 for the bisector span and 36.7 and 16.5 days forthe logR′HK . 36 days periodicity always are detected withFAP < 10−6. These results indicate that activity effects inthe RV at ≈ 36 and ≈ 16 days are to be expected, as wellas low frequency effects. These signals likely stem from therotation period the star, creating signal at the fundamentalfrequency and the first harmonic.

101 102

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HFWHMBISlogRHK

Periodogram of ancillary indicators

101 102

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0.1

0.2

0.3

0.4

0.5

RSS

HFWHMBISlogRHK

Periodogram of ancillary indicators (36 d subtracted)

Fig. B.1: Top: periodograms of the raw ancillary indica-tors time-series (Hα, FWHM, bisector span and logR′HK .We compute, respectively in blue, orange, green and red).Periodogram of the same timeseries when the signal cor-responding to the maximmum peak is injected in the basemodel.

We now turn to the RV, and define the alternative noisemodels we explore. These are Gaussian, with a white com-ponent, as well as an exponential decay and a quasi-periodicterm, as given by the formula

Vkl = δk,l(σ2k + σ2

W ) + σ2R e

− (tk−tl)2

2τ2R +σ2

QP e− (tk−tl)

2

τ2QP

sin2(tk−tlPact

).

(B.1)

where Vkl is the element of the covariance matrix atrow k and column l, δk,l is the Kronecker symbol, σkis the nominal uncertainty on the measurement k, andσW , σR, τR, σQP , Pact are the parameters of our noisemodel. A preliminary analysis on the ancillary indicators(FWHM, S-index, Hα) showed that they all exhibit statis-tically significant variations at ≈ 40 (36 days), as well as aperiodogram peak at 16 d. The 36 and 16 d signals exhibitphases compatible with one another at 1 sigma, except forthe 36 d signal in the FWHM which is 3σ away from thephase fitted on the S-index and Hα. This points to a stel-lar rotation period of 36 d. We consider all the possiblecombinations of values for σR and σW in 0.0,0.5 1.0,1.25,1.5,1.75,2 m/s, τ = 0, 2, 4, 6 d, Pact = 36.5 d, σQP =0,1,2,3,4 m/s and τQP = 18, 36 or 72 d.

The computation of the `1 periodogram is made assum-ing a certain base model. By this, we mean a linear modelwhich is assumed to be by default in the data, and thuswill automatically be fitted. This base model might repre-sent for instance offsets, trends, or certain periodic signals.We try the following base models: one offset, one offset andsmoothed Hα, one offset and smoothed FWHM, or oneoffset and the smoothed Hα and FWHM time-series. Thesmoothing of a given indicator is done via a Gaussian pro-cess. This process has a Gaussian kernel, whose parameters(time-scale and amplitude) have been optimized to max-imise the likelihood of the data.



Appendix B.3: Results

In Fig. B.2, we represent the `1 periodogram obtained withthe noise model with maximum cross validation score withthree different base models (from top to bottom: withoutany indicator, with smoothed Hα and with both smoothedHα and smoothed FWHM). In all cases, we find signals at35-40 d, 15 d and 3.24 d, as well as a peak at 6.55 d. Wemight also find peaks at 21 d, 9.84 d, 2.08 d, 1.92 1.44, or1.21 days, depending on the models used. Note that 2.08and 1.92 d are aliases of each other, so that the presenceof one or the other might originate from the same signal.The model with the overall highest cross validation score in-cludes only the smoothed Hα time-series in the base model,and with the notations of eq. (B.1), σW = 1.75 m/s, σR =1.5 m/s, τ = 2 days, σQP = 0, and corresponds to themiddle figure in Fig. B.2.

We computed the `1-periodogram on a grid from 0 to0.95 cycles per day (we then do not consider periods be-low one day) with all combinations of the base and noisemodels. As in Hara et al. (2020), the models are rankedwith cross validation. We then considered the 20% high-est ranked models (all noise and base models considered),which we denote by CV20, and computed the number oftimes a signal is included in the model. By that, a peak hasa frequency within 1/Tobs of a reference frequency 1/P0,where Tobs is the observation time-span, and has a FAP be-low 0.5. We report these values in Table B.1 for the refer-ence periods P0 corresponding to signals appearing at leastonce in Fig. B.2. Note that, due to the short time-span, thefrequency resolution is not good enough to distinguish 36and 45 days.

We find that signals at 36, 16 and 3.2 days are consis-tently included in the model. The 3.2 days signal presents amedian FAP of 0.002, and can therefore be confidently de-tected. When the base model consists only in an offset, 36and 16 days are systematically significant with FAP <5%,but their significance decreases as activity indicators areincluded in the model. Since these periodicities appear alsoin ancillary indicators, we conclude that they are due toactivity. Signals at 6.5 and 9.9 days appear in ≈ 15% of themodels, but with low significance.

We re-compute the `1 periodogram by assuming that3.2, 6.5, 15.9 and 36.7 days are by default in the data, as wellas the smoothed Hα. Furthermore, the detection of a 6.5 dtransit signal is clear in TESS data only. Both for the 3.2and 6.4 signals, we use the phase predicted from the transitsfor these two planets. This step is done as it improves theability if the `1 periodogram to find the smallest amplitudesignals. We find of signals at 2.08 d (alias of 1.91 d), 1.21,16 d and 9.9 days (see Fig. B.3, top). These are only hints,as the false alarm probabilities of these peaks is above 50%.We note that, depending on the frequency grid chosen, a5.6 d periodicity, alias of 1.21 d, might appear, as shown inFig. B.3, bottom. Although detections cannot be claimedfrom the RV data only, we note that there are signals at2.08 d (alias of 1.91 d) and 9.9 d. We find that they arein phase with the photometric signals within 1.5 σ, whichfurther strengthens the detection of transiting planets atthese periods.

When adding in the base model all known transitingplanets, as well as the smoothed Hα indicators and sinefunctions at 15.7 and 36 days, we obtain B.4, where a signalat 15.2 days appears with a FAP of ≈ 20%.

Table B.1: Inclusion in the 20% best models of differentperiodicities. We report the false alarm probability (FAP)associated with the best model, the frequency of inclusionin the model and the median FAP in the 20% best models.

Period(d)

FAP(best fit)

Inclusionin themodel

CV20 me-dian FAP

1.21 5.33·10−1 1.730% -1.44 1.00 0.0% -1.914 1.00 0.0% -2.08 2.93·10−1 0.384% -3.24 1.33·10−2 100.0% 2.20·10−3

6.5 1.31·10−1 15.57% -9.9 3.41·10−1 16.63% -15.2 8.38·10−2 89.32% 6.83·10−2

20.7 1.00 0.0% -36 (45) 5.24·10−1 95.48% 2.57·10−1

The 20.7 day transiting planet could correspond to thepeak at 21.6 d in the `1 periodogram (Fig. B.2, top). Whenrestricting the frequency grid to 0 to 0.55 cycles per day,the best CV model yields Fig. 7, where a signal at 20.6 daysappear. However, signals close to 20.7 days seems to disap-pear when changing the stellar activity model. The plan-etary signature might be hidden in the RV due to stellareffect. Further observations would allow to better disentan-gle stellar and planetary signals.

Appendix B.4: Detections: conclusion

In conclusion, we can claim an independent detection ofthe 3.2 d planet with RV. We find significant signatures at36 and 15-16 days, which we attribute to activity. We findsignals at 1.91 d or its alias 2.08 d, 6.5 and 9.9 d in phasewith the detected transits. We see a signal at 21 d for somemodels which could correspond to the 20.7 d planet, butit seems that there are not enough points to disentangle aplanetary and an activity signals and this period. We finallynote that modelling an activity signal at 15.7 days leaves asignal at 15.2 days, which likely stems from the presence ofa planet at this period.

Finally, we check the consistency in phase of the signalsfitted onto the photometric and RV data. We perform aMCMC computation of the orbital elements on the RVswith exactly the same priors as model 2, except that we seta flat prior on the phases. We find that the uncertainties onthe phase corresponding to planets b,c,d,e,f,g correspondrespectively to 7, 2, 3,3, 4 and 100% of the period. Thephases from transits are included respectively in the 1.5, 1,2, 1, 1 σ intervals derived from the RVs, such that we deemthe phases derived from RVs consistent with the transits.

Appendix B.5: Mass and density estimates

The RV measurements allow to obtain mass estimates ofthe planets. These ones can depend on the model of stellaractivity used. To account for this, we estimate masses withtwo different stellar activity models: (1) Activity is mod-elled as two sinusoidal signals plus a correlated noise (2)activity is modelled as a correlated Gaussian noise with asemi-periodic kernel. In both cases, we add the Hα time-



101 102 103

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coef

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mpl

itude

36.7

15.93.24

21.61.21 6.551.44


TOI 178, base model: offset, 1 periodogram

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itude

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2.083.10


TOI 178, base model: offset, H , 1 periodogram

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1.6

coef

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mpl

itude

41.2

3.24

14.9

1.44 9.722.083.111.21


TOI 178, base model: offset, H , FWHM, 1 periodogram

Fig. B.2: `1 periodogram of the ESPRESSO RV correspond-ing to the best cross validation score with different linearbase models for the data: Top: only one offset, middle: off-set, smoothed Hα, bottom: smoothed Hα and smoothedFWHM.

series smootheed with a Gaussian process, as describedin B.3. In model (1), the two sinusoids have periods of ≈ 40days and ≈ 16 days. This is motivated by the fact thatthese periodicities appeared systematically in ancillary in-dicators, though with different phases. We therefore allowthe phase to vary freely in the RVs. The priors on the stel-lar activity periods are taken as Gaussians with mean 16days, σ = 1 day and mean 36.8 days, σ = 8 days, accord-ing to the variability of the position of the peaks appearingin the spectroscopic ancillary indicators. We further addfree noise components, a white component and correlatedcomponent with an exponential kernel. The prior on thevariance is a truncated Gaussian with σ = 16 m2/s2, theprior on the noise time scale is log-uniform between 1h and30 days. The second model is a Gaussian process (or here,correlated Gaussian noise), with a quasi-periodic kernel kof the form

k(t;σW , σR, λ, ν) = σ2W + σ2

R e−tτ cos(νt). (B.2)

We impose a Gaussian truncated prior on σ2W , σ

2R with σ =

100 m2/s2. We impose a flat prior on ν between 2π/50and 2π/30 rad/day and a log-uniform prior on τ on 1h to1000 days. For planets b, c, d, g, e, f, the priors on periodand time of conjunction are set according to the constraintsobtained from the joint fit of TESS and CHEOPS data.

We run an adaptive Monte-Carlo Markov chain algo-rithm as described in Delisle et al. (2018) implementingspleaf (Delisle et al. 2020), which offers optimized routines

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itude

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1.2116.1

1.14 9.902.76

3.071.61


TOI 178 1 periodogram

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3.07



Fig. B.3: `1 periodogram of the ESPRESSO RV peri-odogram corresponding to the best cross validation scorewith a base model including Hα, the 6.5 d and 3.2 d plan-ets, a 36 and 15.2 d signals. Top: frequency grid from 0 to0.9 cycles per day, bottom frequency grid from 0 to 0.55cycles per day.

101 102 103

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3.08



Fig. B.4: `1 periodogram when all transiting planets areadded to the base models, as well as a smoothed Hα indi-cator and sinusoids at 15.7 and 36 days.

to compute the inverse of covariance matrices. We check theconvergence of the algorithm by ascertaining that 600 effec-tive samples have been obtained for each variable (Delisleet al. 2018). The posterior medians as well as the 1 σcredible intervals are reported in Table B.2 for both ac-tivity models. The kernel (B.2) is close to the Stochasticharmonic oscillator (SHO), as defined in Foreman-Mackeyet al. (2017b). We also tried the SHO kernel, and simplyimposed that the quality factor Q be greater than one half,the other parameters having a flat priors. We find very sim-ilar results, so that they are not reported in Table B.2.

Modelling errors might leave a trace in the residuals.In Hara et al. (2019), it is show that, if the model used forthe analysis is correct, the residuals of the maximum likeli-hood model appropriately weighted should follow a normaldistribution and not exhibit correlations. In Fig. B.5, weshow the histogram of the residuals (in blue) and a the



Table B.2: mass estimation with activity model as two si-nusoids, Hα model and correlated noise

Parameter Estimates (1) Estimates (2)Planets

TOI 178b (1.9 days)K [m/s] 1.07+0.17

−0.15 1.04+0.27−0.28

m [M⊕] 1.54+0.28−0.25 1.49+0.40

−0.43

ρ [ρ⊕] 0.93+0.20−0.25 0.90+0.28

−0.31

TOI 178c (3.2 days)K [m/s] 2.82+0.17

−0.17 2.72+0.27−0.28

m[M⊕] 4.85+0.46−0.50 4.67+0.62

−0.58

ρ [ρ⊕] 0.92+0.15−0.19 0.89+0.17

−0.19

TOI 178d (6.5 days)K [m/s] 1.43+0.22

−0.22 1.24+0.33−0.29

m[M⊕] 3.11+0.52−0.57 2.70+0.71

−0.71

ρ [ρ⊕] 0.16+0.03−0.03 0.14+0.04

−0.04

TOI 178e (9.9 days)K [m/s] 1.52+0.25

−0.24 1.72+0.31−0.35

m[M⊕] 3.79+0.67−0.70 4.28+0.82

−0.96

ρ [ρ⊕] 0.37+0.07−0.08 0.42+0.09

−0.10

TOI 178f (15.2 days)K [m/s] 2.65+0.34

−0.30 2.88+0.34−0.37

m[M⊕] 7.60+1.05−1.18 8.28+1.11

−1.34

ρ [ρ⊕] 0.56+0.09−0.10 0.61+0.11

−0.11

TOI 178g (20.7 days)K [m/s] 1.41+0.27

−0.28 1.19+0.42−0.47

m [M⊕] 4.51+0.90−1.00 3.78+1.47

−1.45

ρ [ρ⊕] 0.21+0.04−0.04 0.17+0.06

−0.07

Other signals)Signal at ≈ 40 days (probably Prot)

P [days] 39.4+1.09−3 -

K [m/s] 3.01+0.35−0.36 -

Activity signal at 16 daysP [days] 16.2+0.28

−0 -K [m/s] 1.07+0.44

−0.40 -Noise parameters

σW [m/s] 0.62+0.19−0.21 0.90+0.26

−0.25

σR [m/s] 0.47+0.18−0.23 -

τR [days] 5.74+4.03−5.12 -

σQP [m/s] - 2.90+0.99−1.47

τQP [days] - 350+192−337

P [days] - 42.7+3.27−3.72

probability distribution function of a normal variable. Thetwo appear to be in agreement. We further compute theShapiro-Wilk normality test (Shapiro & Wilk 1965) andfind a p-value of 0.78, which is compatible with normality.The variogram does not exhibit signs of correlations. Thesame analysis is performed with model 2, for which theresiduals also do not exhibit non normality nor correlation(p value of 0.99 on the rediduals). We conclude that bothmodels are compatible with the data.

2 1 0 1 2Residuals (m/s)

0.0

0.1

0.2

0.3

0.4

Dens

ity

Normal distributionResiduals

Histogram of the weighted residuals

10 1 100 101 102

Time difference between measurements (d)

4

2

0

2

4

Ampl

itude

(m/s

)

rW(ti) rW(tj)Standard deviation per bin

Variogram of the weighted residuals

Fig. B.5: Histograms of the weighted residuals of the maxi-mum likelihood model (Top) and their variogram (bottom)for activity modelled as two sinusoidal functions (model 1).

Appendix C: Continuation of a Laplace resonantchain

TOI-178 is in a configuration where successive pair of planetis at the same distance to the exact neighbouring first orderMMR. Generalising a bit the configuration, we define fictiveplanets 1, 2 and 3, such that planets 1 and 2 are close to theresonance (k1 +q1) : k1 and planets 2 and 3 are close to theresonance (k2 + q2) : k2, where the ki and qi are integers.we hence write the near-resonant angles:

ϕ1 = k1λ1 − (k1 + q1)λ2 ,

ϕ2 = k2λ2 − (k2 + q2)λ3 ,(C.1)

where λi is the mean longitude of planet i. The associateddistances to the resonances read:

∆1 = k1n1 − (k1 + q1)n2 ,

∆2 = k2n2 − (k2 + q2)n3 ,(C.2)

where ni is the mean motion of planet i. A Laplace relationexists between these three planets if:

j1∆1 − j2∆2 ≈ 0 , (C.3)

where j1 and j2 are integers. In addition, the invariance byrotation of the Laplace angle (D’Alembert relation) gives:

j1q1 = j2q2 . (C.4)

As the result, the Laplace relation requires:

n3 ≈k2n2 − q2

q1∆1

k2 + q2, (C.5)

which translates as, for the period of the 3rd planet:

P3 ≈k2 + q2

k2P2− q2

q1P1,2

, (C.6)



12 13 14 15 16 17 18Pf [day]

0

50

100

150

200

250

300

350

Supe

r per

iod

[d] 2/3 with e

3/4 with e4/5 with e3/2 with g4/3 with g5/4 with g

Fig. C.1: Super-period of planet f with respect to e (inred) and g (in black) for a set of first order MMR. Thepurple line indicate the value of the super-period betweenthe other pairs of the chain 260 day. Only two possibleperiod allowed planet f to be part of the resonant chain,which correspond to the near-intersection of the right-handside of a red curve, left-hand side of a black curve, and thepurple horizontal line.

where P1,2 is the super-period associated to ∆1. From thiswe can compute the periods of potential additional planetsthat would continue the Laplace resonant chain of TOI-178. Taking planets e and f as planets 1 and 2, the formulato compute the possible period of a planet x that couldcontinue the resonant chain becomes:

Px =k + q

kPf− q

Pf,g

, (C.7)

where Pf,g is the super-period between the near first-orderresonances of the known chain defined by eq. (1), here ∼260 days. and k and q are integers such that planet x andf are near a (k + q)/k MMR. Some of the relevant periodsare displayed in table 7.

Similar computation allowed us to determine the possi-ble period of planet f before its confirmation by CHEOPS,see Fig. C.1

Appendix D: Internal structure

We provide here the posterior distributions of the plane-tary interior models, as well as some more details on theircalculations.

As mentioned in the main text, deriving the posteriordistribution of the internal structure parameters requirescomputing millions of times the radius of planets for dif-ferent set of parameters. In order to speed up this calcula-tion, we have first computed a large (five millions points)database of internal structure models, varying the differ-ent parameters. This database was split randomly in threesets, one training set (80% of models), one validation setand one test set (each of them containing 10 % of the wholedatabase). We have then in a second time fitted a Deep Neu-ral Network in order to be able to compute very rapidly the

Fig. D.1: Histogram of prediction error from our DNN onthe test set. The y axis is in log scale, and the x axis coversan error from -1% to 1%.

radius of a planet with a given set of internal structure pa-rameters. The architecture of the DNN we used is made of 6layers of 2048 nodes each, and we used the classical ReLU asactivation function (Alibert & Venturini 2019). The DNNis trained for a few hundreds epochs, using a learning ratethat is progressively reduced from 1.e− 2 down to 1.e− 4.Our DNN allows to compute these radii with an error be-low 0.25% on average with a speed up of many orders ofmagnitude (a few thousands models computed per second).Fig. D.1 shows the prediction error we reach on the test set(which was not used for training). The error on the pre-dicted radius is lower than 0.4%, an error much smallerthan the uncertainty on the radii we obtained for the TOI-178 planets, in 99.9% of the cases. It is finally importantto note that this model does not include the compressioneffect that would be generated by the gas envelope onto thesolid part. Given the mass of the gas envelop in all planets,this approximation is justified.

The following plots shows the posterior distribution ofthe interior structure parameters. The parameters are theinner core, mantle and water mass fraction relative to themass of the solid planet, the Fe, Si, Mg molar fraction inthe mantle, the Fe molar fraction in the core, and the massof gas (log scale). It is important to remember that sincethe core, mantle and water mass fractions add up to one,they are not independent. This is also the case for the Si,Mg and Fe molar fraction in the mantle.



Fig. D.2: Corner plot showing the main parameters of theinternal structure of planet b. The parameters are the coremass fraction, mantle mass fraction, water mass fraction(all relative to the solid planet), molar fractions of Fe, Siand Mg in the mantle, molar fraction of Fe in the core, andmass of gas (log scale). The dashed lines give the positionsof the 16% and 84% quantiles, the number on top of eachcolumn give the median and the same quantiles.

Fig. D.3: Same as Fig. D.2 for planet c.

Fig. D.4: Same as Fig. D.2 for planet d.

Fig. D.5: Same as Fig. D.2 for planet e.



Fig. D.6: Same as Fig. D.2 for planet f.

Fig. D.7: Same as Fig. D.2 for planet g.


six transiting planets and a chain of laplace resonances

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